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Printed by W. DRUGULIN, Leipzig (Germany) 


The translator of this little volume has done me the 
honour to ask me to write a few lines of introduction. And 
1 do this willingly, not only that I may render homage to the 
memory of a friend, prematurely torn from life and from 
science, but also because I am convinced that the work of 
ROBERTO BONOLA deserves all the interest of the studious. 
In it, in fact, the young mathematician will find not only 
a clear exposition of the principles of a theory now classical, 
but also a critical account of the developments which 
led to the foundation of the theory in question. 

It seems to me that this account, although concerned 
with a particular field only, might well serve as a model 
for a history of science, in respect of its accuracy and 
its breadth of information, and, above all, the sound philo 
sophic spirit that permeates it. The various attempts of 
successive writers are all duly rated according to their 
relative importance, and are presented in such a way 
as to bring out the continuity of the progress of science, 
and the mode in which the human mind is led through 
the tangle of partial error to a broader and broader view 
of truth. This progress does not consist only in the ac 
quisition of fresh knowledge, the prominent place is taken 
by the clearing up of ideas which it has involved; and it 
is remarkable with what skill the author of this treatise has 
elucidated the obscure concepts which have at particular 
periods of time presented themselves to the eyes of the 
investigator as obstacles, or causes of confusion. I will 
cite as an example his lucid analysis of the idea of there 

IV Introduction. 

being in the case of Non-Euclidean Geometry, in contrast 
to Euclidean Geometry, an absolute or natural measure of 
geometrical magnitude. 

The admirable simplicity of the author s treatment, 
the elementary character of the constructions he employs, 
the sense of harmony which dominates every part of this 
little work, are in accordance, not only with the artistic 
temperament and broad education of the author, but also 
with the lasting devotion which he bestowed on the Theory 
of Non-Euclidean Geometry from the very beginning of 
his scientific career. May his devotion stimulate others to 
pursue with ideals equally lofty the path of historical and 
philosophical criticism of the principles of science! Such 
efforts may be regarded as the most fitting introduction 
to the study of the high problems of philosophy in general, 
and subsequently of the theory of the understanding, in 
the most genuine and profound signification of the term, 
following the great tradition which was interrupted by the 
romantic movement of the nineteenth century. 

Bologna, October ist } 1911. 

Federigo Enriques. 

Translator s Preface. 

BONOLA S Non-Euclidean Geometry is an elementary 
historical and critical study of the development of that subject. 
Based upon his article inENRiQUES collection of Monographs 
on Questions of Elementary Geometry 1 , in its final form it still 
retains its elementary character, and only in the last chapter 
is a knowledge of more advanced mathematics required. 

Recent changes in the teaching of Elementary Geometry 
in England and America have made it more then ever ne 
cessary that those who are engaged in the training of the 
teachers should be able to tell them something of the 
growth of that science; of the hypothesis on which it 
is built; more especially of that hypotheses on which rests 
EUCLID S theory of parallels; of the long discussion to which 
that theory was subjected; and of the final discovery of the 
logical possibility of the different Non-Euclidean Geometries. 

These questions, and others associated with them, are 
treated in an elementary way in the pages of this book. 

In the English translation, which Professor BONOLA 
kindly permitted me to undertake, I have introduced some 
changes made in the German translation. 2 For permission 
to do so I desire to express my sincere thanks to the firm of 
B. G. TEUBNER and to Professor LIEBMANN. Considerable 
new material has also been placed in my hands by Professor 
BONOLA, including a slightly altered discussion of part of 

1 ENRIQUES, F., Questioni riguardanii la geometria elementare, 
(Bologna, Zanichelli, 1900). 

2 Wissenschaft und Hypothese, IV. Band : Die nichteuklidische 
Geometrie. Historisch-kritische Darstellung ihrer Entwicklung. Von 
R. Bottola. Deutsch v. H. Liebmann. (Teubner, Leipzig, 1908). 

VI Translator s Preface. 

SACCHERI S work, an Appendix on the Independence of Pro- 
jective Geometry from the Parallel Postulate, and some further 
Non-Euclidean Parallel Constructions. 

In dealing with GAUSS S contribution to Non-Euclidean 
Geometry I have made some changes in the original on the 
authority of the most recent discoveries among GAUSS S 
papers. A reference to THIBAUT S proof, and some addit 
ional footnotes have been inserted. Those for which I am 
responsible have been placed within square brackets. I have 
also added another Appendix, containing an elementary 
proof of the impossibility of proving the Parallel Postulate, 
based upon the properties of a system of circles orthogonal 
to a fixed circle. This method offers fewer difficulties than 
the others, and the discussion also establishes some of the 
striking theorems of the hyperbolic Geometry. 

Jt only remains for me to thank Professor GIBSON of 
Glasgow for some valuable suggestions, to acknowledge the 
interest, which both the author and Professor LIEBMANN have 
taken in the progress of the translation, and to express my 
satisfaction that it finds a place in the same collection as 
HILBERT S classical Grundlagen der Geometric. 

P. S. As the book is passing through the press I have 
received the sad news of the death of Professor BONOLA. 
With him the Italian School of Mathematics has lost one of 
its most devoted workers on the Principles of Geometry. 
Professor ENRIQUES, his intimate friend, from whom I heard 
of BONOLA S death, has kindly consented to write a short 
introduction to the present volume. I have to thank him, 
and also Professor W. H. YOUNG, in whose hands, to avoid 
delay, I am leaving the matter of the translation of this 
introduction and its passage through the press. 

The University, Sydney, August 1911. 

H. S. Carslaw. 

Author s Preface. 

The material now available on the origin and develop 
ment of Non-Euclidean Geometry, and the interest felt in 
the critical and historical exposition of the principles of the 
various sciences, have led me to expand the first part of my 
article Sulla teoria delle parallde e suite geometric non- 
euclidee which appeared six years ago in the Question! ri- 
guardanti la geometria elementare, collected and arranged 
by Professor F. ENRIQUES. 

That article, which has been completely rewritten for the 
German translation T of the work, was chiefly concerned with 
the systematic part of the subject. This book is devoted, on 
the other hand, to a fuller treatment of the history of parallels, 
and to the historical development of the geometries of Lo- 


In Chapter I. , which goes back to the work of EUCLID 
and the earliest commentators on the Fifth Postulate, I have 
given the most important arguments, by means of which 
the Greeks, the Arabs and the geometers of the Renaissance 
attempted to place the theory of parallels on a firmer 
foundation. In Chapter II, relying chiefly upon the work of 
SACCHERI, LAMBERT and LEGENDRE, I have tried to throw 
some light on the transition from the old to the new 
ideas, which became prevalent in the beginning of the i gth 
Century. In Chapters III. and IV., by the aid of the in- 

i ENRIQUES, F., Fragen der Elementargeometrie. I. Teil: Die 
Grundlagen der Geometric. Deutsch von H. THIEME. (1910.) 
II. Teil: Die geometrischen Aufgaben, ihre Losung und Losbarkeit. 
Deutsch von H. FLEISCHER. (1907.) Teubner, Leipzig. 

vni Author s Preface. 

vestigations of GAUSS, SCHWEIKART, TAURINUS, and the con 
structive work of LOBATSCHEWSKY and BOLYAI, I have ex 
plained the principles of the first of the geometrical systems, 
founded upon the denial of EUCLID S Fifth Hypothesis. In 
Chapter V., I have described synthetically the further deve 
lopment of Non-Euclidean Geometry, due to the work of 
RIEMANN and HELMHOLTZ on the structure of space, and 
to CAYLEY S projective interpretation of the metrical proper 
ties of geometry. 

In the whole of the book I have endeavoured to pre 
sent, the various arguments in their historical order. How 
ever when such an order would have made it impossible to 
treat the subject simply, I have not hesitated to sacrifice it, 
so that I might preserve the strictly elementary character of 
the book. 

Among the numerous postulates equivalent to EUCLID S 
Fifth Postulate, the most remarkable of which are brought 
together at the end of Chapter IV., there is one of a statical 
nature, whose experimental verification would furnish an 
empirical foundation of the theory of parallels. In this we 
have an important link between Geometry and Statics 
(GENOCCHI); and as it was impossible to find a suitable place 
for it in the preceding Chapters, the first of the two Notes x 
in the Appendix is devoted to it. 

The second Note refers to a theory no less interesting. 
The investigations of GAUSS, LOBATSCHEWSKY and BOLYAI on 
the theory of parallels depend upon an extension of one of 
the fundamental conceptions of classical geometry. But a 
conception can generally be extended in various directions. 
In this case, the ordinary idea of parallelism, founded on 
the hypothesis of non-intersecting straight lines, coplanar and 

* In the English translation these Notes are called Appendix I. 
and Appendix II. 

Author s Preface. IX 

equidistant, was extended by the above-mentioned geometers, 
who gave up EUCLID S Fifth Postulate (equidistance), and 
later, by CLIFFORD, who abandoned the hypothesis that the 
lines should be in the same plane. 

No elementary treatment of CLIFFORD S parallels is avail 
able, as they have been studied first by the projective 
method (CLIFFORD-KLEIN) and later, by the aid of Different 
ial-Geometry (BiANCHi-FuBiNi). For this reason the second 
Note is chiefly devoted to the exposition of their simplest 
and neatest properties in an elementary and synthetical 
manner. This Note concludes with a rapid sketch of CLIF 
FORD-KLEIN S problem, which is allied historically to the 
parallelism of CLIFFORD. In this problem an attempt is made 
to characterize the geometrical structure of space, by assum 
ing as a foundation the smallest possible number of postul 
ates, consistent with the experimental data, and with the 
principle of the homogeneity of space. 

This is, briefly, the nature of the book. Before sub 
mitting the little work to the favourable judgment of its 
readers, I wish most heartily to thank my respected teacher, 
Professor FEDERIGO ENRIQUES, for the valuable advice with 
which he has assisted me in the disposition of the material 
and in the critical part of the work; Professor CORRADO SEGRE, 
for kindly placing at my disposal the manuscript of a course 
of lectures on Non-Euclidean geometry, given by him, three 
years ago, in the University of Turin; and my friend, Professor 
GIOVANNI VAILATI, for the valuable references which he has 
given me on Greek geometry, and for his help in the cor 
rection of the proofs. 

Finally my grateful thanks are due to my publisher 
CESARE ZANICHELLI, who has so readily placed my book in 
his collection of scientific works. 

Pavia, March, 1906. 

Roberto Bonola. 

Table of Contents. 

Chapter I. pageg 

The Attempts to prove Euclid s Parallel Postulate. 

S 15- The Greek Geometers and the Parallel 

Postulate i g 

S 6. The Arabs and the Parallel Postulate .. .. 9 12 
S 7io. The Parallel Postulate during the Renais 
sance and the 17^ Century 12 21 

Chapter II. 

The Forerunners of Non-Euclidean Geometry. 

S 1117. GEROLAMO SACCHERI (16671733) .. .. 2244 

S 18 22. JOHANN HEINRICH LAMBERT (1728 1777) 4451 
S 2326. The French Geometers towards the End 

of the l8*h Century 51 55 

S 27 28. ADRIEN MARIE LEGENDRE (17521833) .. 55 60 

S 29. WOLFGANG BOLYAI (17751856) 6062 

S 30. FRIEDRICH LUDWIG WACHTER (1792 1817) .. 6263 

S 30 (bis) BERNHARD FRIEDRICII THIB\UT (17761832) 63 

Chapter III. 

The Founders of Non-Euclidean Geometry. 

S 3 1 34- KARL FRIEDRICH GAUSS (17771855) .. 64 75 

S 35. FERDINAND KARL SCHWEIKART (17801859) .. 7577 

S 3638. FRANZ ADOLF TAURINUS (17941874) .. 7783 

Chapter IV. 

The Founders of Non-Euclidean Geometry (Cont). 


(17931856) , 8496 

S 4655- JOHANN BOLYAI (18021860) 96113 

S 5658. The Absolute Trigonometry 113 118 

S 59- Hypotheses equivalent to Euclid s Postulate .. 118 121 

S 60 65. The Spread of Non-Euclidean Geometry 121128 

Chapter V. 

The Later Development of Non-Euclidean Geometry. 

S 66. Introduction 129 

Table of Contents. XI 

Differential Geometry and Xon-Eucliclean Geometry. 

S 6769. Geometry upon a Surface 130139 

7076. Principles of Plane Geometry on the Ideas 

of RIEMANN 139150 

77. Principles of RIEMANN S Solid Geometry.. .. 151 152 

S 78. The Work of HELMHOLTZ and the Investigations 

of LIE I5 2 154 

Projective Geometry and Non-Euclidean Geometry. 

S 79 83. Subordination of Metrical Geometry to 

Projective Geometry 154 164 

84 91. Representation of the Geometry of LOBAT- 

SCHEWSKV-BOLYAI on the Euclidean Plane .. .. 164 175 

S 92. Representation of RIEMANN S Elliptic Geometry 

in Euclidean Space 175 176 

93. Foundation of Geometry upon Descriptive Pro 
perties 176 177 

S 94. The Impossibility of proving Euclid s Postulate 177 *8o 

Appendix I. 

The Fundamental Principles of Statics and Euclid s 

S 13. On the Principle of the Lever 181 184 

S 48. On the Composition of Forces acting at 

a Point 184-192 

S 910. Non-Euclidean Statics 192195 

S II 12. Deduction of Plane Trigonometry from 

Statics 195 199 

Appendix II. 

CLIFFORD S Parallels and Surface. Sketch of CLIFFORD- 
KLEIN S Problem. 

S i 4. CLIFFORD S Parallels 200206 

S 58. CLIFFORD S Surface 206211 

S 9 11. Sketch of CLIFFORD-KLEIN S Problem .. 211-215 
Appendix III. 
The Non-Euclidean Parallel Construction and other 

Allied Constructions. 

S 13- The Non-Euclidean Parallel Construction .. 216 222 
S 4. Construction of the Common Perpendicular to 

two non-intersecting Straight Lines 222223 

S 5. Construction of the Common Parallel to the 

Straight Lines which bound an Angle 223 224 

Xn Table of Contents. 


S 6. Construction of the Straight Line which is per 
pendicular to one of the lines bounding an acute 

Angle and Parallel to the other 224 

S 7. The Absolute and the Parallel Construction .. 224226 

Appendix IV. 

The Independence of Projective Geometry from Euclid s 

S i. Statement of the Problem .. 227228 

S 2. Improper Points and the Complete Projective 

Plane , 228-229 

S 3. The Complete Projective Line .. .. 229 

S 4. Combination of Elements 229 231 

S 5. Improper Lines 231233 

S 6. Complete Projective Space 233 

S 7. Indirect Proof of the Independence of Pro 
jective Geometry from the Fifth Postulate .. 233234 
S 8. BELTRAMI S Direct Proof of this Independence 234236 
S 9. KLEIN S Direct Proof of this Independence .. 236237 
Appendix V. 

The Impossibility of proving Euclid s Postulate. 
An Elementary Demonstration of this Impossibility 
founded upon the Properties of the System of 
Circles orthogonal to a Fixed Circle. 

S i. Introduction 238 

S 27. The System of Circles passing through a 

Fixed Point 239 250 

S 812. The System of Circles orthogonal to a 

Fixed Circle - 250264 

Index of Authors 265 

Chapter I. 

The Attempts to prove Euclid s Parallel 

The Greek Geometers and the Parallel Postulate. 

i. EUCLID (circa 330275, B. C.) calls two straight 
lines parallel, when they are in the same plane and being 
produced indefinitely in both directions, do not meet one 
another in either direction (Def. XXIII.). 1 He proves that 
two straight lines are parallel, when they form with one of 
their transversals equal interior alternate angles, or equal 
corresponding angles, or interior angles on the same side 
which are supplementary. To prove the converse of these 
propositions he makes use of the following Postulate (V.): 

If a straight line falling on two straight lines make the 
interior angles on the same side less than two right angles, 
the two straight lines > if produced indefinitely, meet on that 
side on which are the angles less than the tiuo right angles. 

The Euclidean Theory of Parallels is then completed 
by the following theorems: 

Straight lines which are parallel to the same straight 
line are parallel to each other (Bk. L, Prop. 30). 

i With regard to EUCLID S text, references are made to the 
critical edition of J. L. HEIBERG (Leipzig, Teubner, 1883). [The 
wording of this definition (XXIII), and of Postulate V below, are 
taken from Heath s translation of HEIBRRG S text. (Camb. Univ. Press, 



2 I. The Attempts to prove Euclid s Parallel Postulate. 

Through a given point one and only one straight line 
can be drawn which will be parallel to a given straight line 
(Bk. I. Prop. 31). 

The straight lines joining the extremities of two equal 
and parallel straight lines are equal and parallel (Bk. I. 

Prop- 33)- 

From the last theorem it can be shown that two parallel 
straight lines are equidistant from each other. Among the 
most noteworthy consequences of the Euclidean theory are 
the well-known theorem on the sum of the angles of a tri 
angle, and the properties of similar figures. 

2. Even the earliest commentators on EUCLID S text 
held that Postulate V. was not sufficiently evident to be 
accepted without proof, and they attempted to deduce it as 
a consequence of other propositions. To carry out their pur 
pose, they frequently substituted other definitions of parallels 
for the Euclidean definition, given verbally in a negative 
form. These alternative definitions do not appear in this 
form, which was believed to be a defect. 

PROCLUS (410 485) in his Commentary on the First 
Book of Euclid f hands down to us valuable informa 
tion upon the first attempts made in this direction. He states, 
for example, that POSIDONIUS (i st Century, B. C.) had pro 
posed to call two equidistant and coplanar straight lines par 
allels. However, this definition and the Euclidean one 
correspond to two facts, which can appear separately, and 

* "When the text of PROCLUS is quoted, we refer to the edi 
tion of G. FRLEDLEIN: Prodi Diadochi in primum Euclidis element- 
orum librum commentarii, [Leipzig, Teubner, 1873). [Compare also 
W. B. FRANKLAND, The First Book of Euclid s Elements with a 
Commentary based principally upon that of Proclus Diadochus, (Camb. 
Univ. Press, 1905). Also HEATH S Euclid, Vol. I., Introduction, 
Chapter IV., to which most important work reference has been 
made on p. i]. 

The Commentary of Proclus. 3 

PROCLUS (p. 177), referring to a work by GEMINUS (i st Cen 
tury, B. C.), brings forward in this connection the examples 
of the hyperbola and the conchoid, and their behaviour with 
respect to their asymptotes, to show that there might be 
parallel lines in the Euclidean sense, (that is, lines which 
produced indefinitely do not meet), which would not be 
parallel in the sense of POSIDONIUS, (that is, equidistant). 

Such a fact is regarded by GEMINUS, quoting still from 
PROCLUS, as the most paradoxical [rrapaboHoTaTOv] in the 
whole of Geometry. 

Before we can bring EUCLID S definition into line 
with that of POSIDONIUS, it is necessary to prove that if two 
coplanar straight lines do not meet, they are equidistant; or, 
that the locus of points, which are equidistant from a straight 
line, is a straight line. And for the proof of this proposition 
EUCLID requires his Parallel Postulate. 

However PROCLUS (p. 364) refuses to count it among 
the postulates. In justification of his opinion he remarks 
that its converse ( The sum of two angles of a triangle is less 
than two right angles), is one of the theorems proved by 
EUCLID (Bk. I. Prop. 17); 

and he thinks it impossible / 

that a theorem whose con- A /p 

verse can be proved, is not 
itself capable of proof. Also 
he utters a warning against Q 
mistaken appeals to self- 
evidence, and insists upon 
the (hypothetical) possibi 
lity of straight lines which Fi g- 
are asymptotic (p. 191 2). 

PTOLEMY (2 nd Century, A. D.) we quote again from 
PROCLUS (p. 362 5) attempted to settle the question by 
means of the following curious piece of reasoning. 




4 I. The Attempts to prove Euclid s Parallel Postulate. 

Let AB, CD, be two parallel straight lines and FG a 
transversal (Fig. i). 

Let a, p be the two interior angles to the left of FG, 
and a , p the two interior angles to the right. 

Then a -f- p will be either greater than, equal to, or less 
than a -f- P . 

It is assumed that if any one of these cases holds for 
one pair of parallels (e. g. a + p ^> 2 right angles) this case 
will also hold for every other pair. 

Now FB, GD, are parallels; as are also FA and GC. 

Since a -f P > 2 right angles, 

it follows that a + p > 2 right angles. 

Thus a-r-p + a +P >4 right angles, 

which is obviously absurd. 
Hence a + p cannot be greater than 2 right angles. 

In the same way it can be shown that 

a -f p cannot be less than 2 right angles. 

Therefore we must have 

a -f P = 2 right angles (PROCLUS, p. 365). 
From this result EUCLID S Postulate can be easily obtained. 

3. PROCLUS (p. 371), after a criticism of Ptolemy s 
reasoning, attempts to reach the same goal by another path. 
His demonstration rests upon the following proposition, 
which he assumes as evident: The distance between two 
points upon two intersecting straight lines can be made as great 
as we please, by prolonging the two lines sufficiently. 1 - 

From this he deduces the lemma: A straight line which 
meets one of two parallels must also meet the other. 

1 For the truth of this proposition, which he assumes as self- 
evident, PROCLUS relies upon the authority of ARISTOTLE. Cf. 
De Coelo I., 5. A rigorous demonstration of this very theorem 
was given by SACCHERI in the work quoted on p. 22. 

Proclus (continued). tj 

His proof of this lemma is as follows : 
Let AB, CD, be two parallels and EG a transversal, 
cutting the former in F (Fig. 2). 

Fig. 2. 

The distance of a variable point on the ray FG from 
the line AB increases without limit, when the distance of that 
point from F is increased indefinitely. But since the distance 
between the two parallels is finite, the straight line EG must 
necessarily meet CD. 

PROCLUS, however, introduced the hypothesis that the 
distance between two parallels remains finite; and from this 
hypothesis EUCLID S Parallel Postulate can be logically de 

4. Further evidence of the discussion and research 
among the Greeks regarding Euclid s Postulate is given by 
the following paradoxical argument. Relying upon it, accord 
ing to PROCLUS, some held that it had been shown that two 
straight lines, which are cut by a third, do not meet one 
another, even when the sum of the interior angles on the 
same side is less than two right angles. 

Let AC be a transversal of the two straight lines AB, 
CD and let E be the middle point of AC (Fig. 3). 

On the side of A C on which the sum of the two internal 
angles is less than two right angles, take the segments AF 
and CG upon AB and CD each equal to AE. The two 
lines AB and CD cannot meet between the points AF and 
CG, since in any triangle each side is less than the sum of 
the other two. 

6 I. The Attempts to prove Euclid s Parallel Postulate. 

The points F and G are then joined, and the same 
process is repeated, starting from the line FG. The segments 
FK and GL are now taken on AB and CD, each equal to 
half of FG. The two lines AB, CD are not able to meet 
between the points F t K and G, L. 

Since this operation 
can be repeated indefini 
tely, it is inferred that the 
two lines AB, CD will never 

The fallacy in this ar 
gument is contained in the 
use of infinity, since the 
segments AF, FK could 
tend to zero, while their 

sum might remain finite. The author of this paradox has 
made use of the principle by means of which ZENO (495 
435 B. C.) maintained that it could be proved that Achilles 
would never overtake the tortoise, though he were to travel 
with double its velocity. 

This is pointed out, under another form, by PROCLUS 
(p. 369 70), where he says that this argument proves that 
the point of intersection of the lines could not be reached 
(to determine, 6pieiv) by this process. It does not prove 
that such a point does not exist. 1 

Proclus remarks further that since the sum of two 
angles of a triangle is less than two right angles (Eucuo Bk. I. 
Prop. 17), there exist some lines, intersected by a third, 
which meet on that side on which the sum of the interior 

1 [Suppose we start with a triangle ABC and bisect the base 
BC in D. Then on BA take the segment BE equal to BD, and 
on CA [the segment CF equal to CD, and join EF, Then repeat 
this process indefinitely. The vertex A can never be reached by 
this means, although it is at a finite distance.] 

Proclus (continued . n 

angles is less than two right angles. Thus if it is asserted 
that for every difference between this sum and two right 
angles the lines do not meet, it can be replied that for 
greater differences the lines intersect. 

But if there exists a point of section, for certain pairs 
of lines, forming with a third interior angles on the same 
side whose sum is less than two right angles, it remains to be 
shown that this is the case for all the pairs of lines. Since 
it might be urged that there could be a certain deficiency (from 
two right angles) for which they (the lines) would not inter 
sect, while on the other hand all the other lines, for which the 
deficiency was greater, would intersect! (PROCLUS, p. 371.) 

From the sequel it will appear that the question, which 
Proclus here suggests, can be answered in the affirmative 
only in the case when the segment AC of the transversal 
remains unaltered, while the lines rotate about the points A 
and C and cause the difference from two right angles to vary. 

5. Another very old proof of the Fifth Postulate, 
reproduced in the Arabian Commentary of AL-NiRizi 1 (9^ 
Century), has come down to us through the Latin translation 
ofGHERARDO DA CREMONA (i 2*h Century), and is attributed 
to AcANis.3 

The part of this commentary relating to the definitions, 
postulates and axioms, contains frequent references to the 

1 Cf. R. O. BESTHORN u. J. L. HEIBERG, Codex Leidensis, 
399, /. Euclidis Elementa ex inter pretatione Al-Hadschdschadsch cum 
commentaries Al~Narizii y (Copenhagen, F. Hegel, 189397). 

2 Cf. M. CURTZE, Anaritii in decent libros priores clementorum 
Euclidis Commentarii. Ex interpretatione Gherardi Cremonensis in 
Codice Cracoviensi 569 servata, (Leipzig, Teubner, 1899). 

3 With regard to AGANIS it is right to mention that he is 
identified by CURTZE and HEIBERG with GEMINUS. On the other 
hand P. TANNERY does not accept this identification. Cf. TANNERY, 
*Lc philosophe Aganis est-il identique d Gf minus? Bibliotheca Math. 
<3) B.d. II. p. 9 u [1901]. 

8 I. The Attempts to prove Euclid s Parallel Postulate. 

the name of Sambelichius, easily identified with Simplicius, 
the celebrated commentator on Aristotle, who lived in the 
6 th Century. It would thus appear that Simplicius had written 
an Introduction to the First Book of Euclid, in which he ex 
pressed ideas similar to those of Geminus and Posidonius, 
affirming that the Fifth Postulate is not self evident, and 
bringing forward the demonstration of his friend AGANIS. 

This demonstration is founded upon the hypothesis that 
equidistant straight lines exist. AGANIS calls these parallels, 
as had already been done by Posidonius. From this hypo 
thesis he deduces that the shortest distance between two 
parallels is the common perpendicular to both the lines: 
that two straight lines perpendicular to a third are parallel 
to each other: that two parallels, cut by a third line, form 
interior angles on the same side, which are supplementary, 
and conversely. 

These propositions can be proved so easily that it is 
unnecessary for us to reproduce the reasoning of AGANIS. 
Having remarked that Propositions 30 and 33 of the First 
Book of EUCLID follow from them, we proceed to show how 
AGANIS constructs the point of intersection of two straight 
lines which are not equidistant. 

Let AB, GD be two straight lines cut by the trans 
versal EZ, and such that the sum of the interior angles AEZ^ 
EZD is less than two right angles (Fig. 4). 

Without making our figure any less general we may sup 
pose that the angle AEZ is a right angle. 

Upon ZD take an arbitrary point T. 

From T draw TL perpendicular to ZE. 

Bisect the segment EZ at P\ then bisect the segment 
PZ at M\ and then bisect the segments MZ, etc. . . . until 
one of the middle points P, M, . . . falls on the segment LZ. 

Let this point, for example, be the point M. 

Draw MN perpendicular to EZ, meeting ZD in N. 

Equidistant Straight Lines. O 

Finally from ZD cut off the segment ZC, the same 
multiple of Z/V as ZE is of ZM. 

In the case taken in the figure ZC = 4 ZN. 

The point C thus obtained is the point of intersection of 
the two straight lines AB and GD. 
C F 



To prove this it would be necessary to show that the 
equal segments ZN, JVS, . . ., which have been cut off one 
after the other from the line ZD, have equal projections on 
ZE. We do not discuss this point, as we must return to it 
later (p. n). In any case the reasoning is suggested directly 
by AGANIS figure. 

The distinctive feature of the preceding construction is 
to be noticed. It rests upon the (implicit) use of the so-called 
Postulate of Archimedes, which is necessary for the deter 
mination of the segment J/Z, less than LZ and a submult- 
iple of EZ. 

The Arabs and the Parallel Postulate. 

6. The Arabs, succeeding the Greeks as leaders in 
mathematical discovery, like them also investigated the Fifth 

Some, however, accepted without hesitation the ideas 
and demonstrations of their teachers. Among this number is 
AL-NIRIZI (9 th Century), whose commentary on the definitions, 

IO ! The Attempts to prove Euclid s Parallel Postulate. 

postulates and axioms ot the First Book is modelled on the 
Introduction to the Elements of SIMPLICIUS, while his demon 
stration of the Fifth Euclidean Hypothesis is that of AGANIS, to 
which we have above referred. 

Others brought their own personal contribution to the 
argument. NASIR-EDDIN [1201 1274], for example, although 
in his proof of the Fifth Postulate he employs the criterion 
used by Aganis, deserves to be mentioned for his original idea 
of explicitly putting in the forefront the theorem on the sum 
of the angles of a triangle, and for the exhaustive nature of 
his reasoning. 1 

The essential part of his hypothesis is as follows : If two 
straight lines r and s are the one perpendicular and the other 
oblique to the segment AB, the perpendiculars drawn from s 
upon r are less than AB on the side on which s makes an acute 
angle with AB, and greater on the side on which s makes an 
obtuse angle with AB. 

It follows immediately that if AB and A 1? are two equal 
perpendiculars to the line BB from the same side, the line 
AA is itself perpendicular to both AB and A ff. Further 
we have A A = BB ; and therefore the figure AA&B is a 
quadrilateral with its angles right angles and its opposite sides 
equal, i. e., a rectangle. 

From this result NASIR-EDDIN easily deduced that the sum 
of the angles of a triangle is equal to two right angles. For 
the right-angled triangle the theorem is obvious, as it is half 
of a rectangle; for any triangle we obtain it by breaking up 
the triangle into two right-angled triangles. 

With this introduction, we can now explain shortly how 
the Arabian geometer proves the Euclidean Postulate [cf. 

1 Cf. : Euclidis elementorum libri XII studii Nassiredini % (Rome, 
1594). This work, written in Arabic, was republished in 1657 and 
1801. It has not been translated into any other language. 

Nasir-Eddin s Proof. 








Fig. 5- 

Let AB, CD be two rays, the one oblique and the other 
perpendicular to the straight line AC (Fig. 5). From AB cut 
off the part AH, and from H draw the perpendicular HH 
to AC. If the point H falls on C, or on the opposite side 
of C from A, the two rays AB and 
CD must intersect. If, however, H 
falls between A and C, draw the line 
AL perpendicular to AC and equal 
to HH . Then, from what we have 
said above, HL = AH . In Affpro- 
<iuced take HK equal to AH. From 
K draw KK perpendicular to AC. 
Since KK > HH , we can take 
K L = H H, and join L H. The 
quadrilaterals K H HL , H ALHaxt both rectangles. There 
fore the three points Z , H, L are in one straight line. It fol 
lows that <$:L HK=^AHL, and that the triangles AHL, 
JIL K axt equal. Thus L H= HL, and from the properties 
of rectangles, K H = HA. 

In HK produced, take KM equal to KH. From M 
draw MM perpendicular to AC. By reasoning similar to 
what has just been given, it follows that 

M K = K H = H A. 

This result obtained, we take a multiple of AH greater 
than AC {The Postulate of Archimedes}. For example, let 
AO , equal to 4 AH , be greater than AC. Then from AB 
cut r off AO= $AH, and draw the perpendicular from O 
to AC. 

This perpendicular will evidently be OO . Then, in the 
right-angled triangle AO O, the line CD, which is perpendicu 
lar to the side AO , cannot meet the other side OO\ and it 
must therefore meet the hypotenuse OA. 

By this means it has been proved that two straight lines 
AB, CD, must intersect, when one is perpendicular to the 

12 I. The Attempts to prove Euclid s Parallel Postulate. 

transversal AC and the other oblique to it. In other words 
the Euclidean Postulate has been proved for the case in which 
one of the internal angles is a right angle. 

NASIR-EDDIN now makes use of the theorem on the sum 
of the angles of a triangle, and by its means reduces the 
general case to this particular one. We do not give his reas 
oning, as we shall have to describe what is equivalent to 
it in a later article, [cf. p. 37.] l 

The Parallel Postulate during the Renaissance and 
the 17 th Century. 

7. The first versions of the Elements made in the 
1 2th an d j^th Centuries on the Arabian texts, and the later 
ones, made at the end of the 1 5 th and the beginning of the 
1 6 th , based on the Greek texts, contain hardly any critical 
notes on the Fifth Postulate. Such criticism appears after the 
year 1550, chiefly under the influence of the Commentary of 
Proclus? To follow this more easily we give a short sketch 
of the views taken by the most noteworthy commentators of 
the 1 6th and jyth centuries. 

F. COMMANDINO [1509 157$] adds to the Euclidean 
definition of parallels, without giving any justification for this 

1 NASIR-EDDIN S demonstration of the Fifth Postulate is given 
in full by the English Geometer J. WALLIS, in Vol. II. of his works 
(cf. Note on p. 15^, and by G. CASTILLON, in a paper published in 
the Mem. de 1 Acad. roy. de Sciences et Belles-Lettres of Berlin, 
T. XVIII. p. 175183, (17881789). In addition, several other 
writers refer to it, among whom we would mention chiefly, G. S. 
KLllGEL, (cf. note, (3), p. 44), J. HOFFMAN, Kritik der Parallelentheorie, 
(Jena, 1807); V. FLAUTI, Nttova dimostrazione del postulate quinto, (Na 
ples, 1818). 

2 The Commentary of Proclus was first printed at Basle (1533) 
in the original text; and next at Padua (1560) in Barozzi s Latin 

Italian Mathematicians of the Renaissance. 13 

step, the idea of equidistance. With regard to the Fifth Postul 
ate he gives the views and the demonstration of PROCLUS. 1 

C. S. CLAVIO [1537 1612], in his Latin translation of 
Euclid s text 2 , reproduces and criticises the demonstration of 
PROCLUS. Then he brings forward a new demonstration of the 
Euclidean hypothesis, based on the theorem: The line equi 
distant from a straight line is a straight line; which he at 
tempts to justify by similar reasoning. His demonstration 
has many points in common with that of Nastr-Eddin. 

P. A. CATALDI [? 1626] is the first modern mathema 
tician to publish a work devoted exclusively to the theory of 
parallels. 3 CATALDI starts from the conception of equidistant 
and non-equidistant straight lines; but to prove the effective 
existence of equidistant straight lines, he adopts the hypothesis 
that straight lines which are not equidistant converge in one 
direction and diverge in the other, [cf. NASIR-EDDIN.] *. 

G. A. BORELLI [1608 1679] takes the following Axiom 
| XIV], and attempts to justify his assumption: 

l lf a straight line which remains always in the same plane 
as a second straight line, moves so that the one end always touches 
this line, and during the whole displacement the first remains 
continually perpendicular to the second, then the other end, as it 
moves, will describe a straight line! 

Then he shows that two straight lines which are perpen 
dicular to a third are equidistant, and he defines parallels as 
equidistant straight lines. 

The theory of parallels follows. 5 

1 Elementorum libri XV, (Pesaro, 1572). 

2 Eudidis elementorum libri XV, (Rome, 1574). 

3 Operetta delle linee rette equidistanti et non equidistanti t (Bologna, 
I6o 3 ). 

4 CATALDI made some further additions to his argument in the 
work, Aggiunta all operetta delle linee rette equidistanti ft non equi 
distanti. (Bologna, 1604). 

5 BORELLI: Eudides restitutus, (Pisa, 1658). 

14 I. The Attempts to prove Euclid s Parallel Postulate. 

8. GIORDANO VITALE [1633 1711] again returns to 
the idea of equidistance put forward by POSLDONIUS, and re 
cognizes, with PROCLUS, that it is necessary to exclude the pos 
sibility of the Euclidean parallels being asymptotic lines. To 
this end he defines two equidistant straight lines as parallels, 
and attempts to prove that the locus of the points equidistant 
from one straight line is another straight line. 1 

His demonstration practically depends upon the follow 
ing lemma: 

If two points, A, C upon a curve, whose concavity is to 
wards X, are joined by the straight line AC, and perpendiculars 
are drawn from the infinite number of points of the arc AC 
upon any straight line, then these perpendiculars cannot be equal 
to each other. 

The words any straight line , in this enunciation, do not 

refer to a straight line taken at random in the plane, but to 

p a straight line constructed in 

the following way (Fig. 6). 

From the point B of the arc 

.--"" I --^^ AC draw BD perpendicular to 

A C the chord^C. Then at A draw 

Flgl 6< AG also perpendicular to AC. 

Finally, having cut off equal segments AG and DF upon 
these two perpendiculars, join the ends G and F. GF is the 
straight line which GIORDANO considers in his demonstration, 
a straight line with respect to which the arc AB is certainly 
not an equidistant line. 

But when the author wishes to prove that the locus of 
points equidistant from a straight line is also a straight line, 
he applies the preceding lemma to a figure in which the re 
lations existing between the arc ABC and the straight line 

1 GIORDANO VITALE: Euclide restitute overo gli antichi dementi 
geometrici ristaurati, e fadlitati. Libri XV. (Rome, 1680). 

Giordano Vitale s Proof. j ^ 

GF do not hold. Thus the consequences which he deduces 
from the existence of equidistant straight lines are not really 

From this point of view GIORDANO S proof makes no ad 
vance upon those which preceded it. However it includes a 
most remarkable theorem, containing an idea which will be 
further developed in the articles which follow. 

Let ABCD be a quadrilateral of which the angles A, B 
are right angles and the sides AD, BC ^ H 
equal (Fig. 7). Further, \n\.HK be the per 
pendicular drawn from a point H, upon the 
side DC, to the base AB of the quadri 
lateral. GIORDANO proves: (i) that the ang 
les D, C are equal; (ii) that, when the seg 
ment HK is equal to the segment AD^ the 
two angles D, C are right angles, and CD is equidistant 
from AB. 

By means of this theorem GIORDANO reduces the question 
of equidistant straight lines to the proof of the existence of 
one point ^Tupon DC, whose distance from AB is equal to 
the segments AD and BC. We regard this as one of the 
most noteworthy results in the theory of parallels obtained 
up to that date. 1 

9. J. WALLIS [1616 1703] abandoned the idea of 
equidistance, employed without success by the preceding 
mathematicians, and gave a new demonstration of the Fifth 
Postulate. He based his proof on the Axiom: To every figure 
there exists a similar figure of arbitrary magnitude. We now 
describe shortly how WALLIS proceeds:* 

1 Cf. : BoNOLA : Un teorema dl Giordano Vitale da Bitonto sulle 
rette cquidistanti, Bollettino di Bibliografia e Storia delle Scienze 
Mat. (1905). 

* Cf.: WALLIS: De Postulate Quinto; et Definizionc Quinta; Lib. 6. 

1 6 I The Attempts to prove Euclid s Parallel Postulate. 

Let a, b be two straight lines intersected at A, B by the 
transversal c (Fig. 8). Let a, p be the interior angles on the 
, h h h same side of c, such that a + (3 is 

less than two right angles. Through 
A draw the straight line b so that 
b and b form with c equal corre 
sponding angles. It is clear that 
b will lie in the angle adjacent to 

f a. Let the line b be now moved 

A w n 

c i continuously along the segment 

Fig> 8 AB, so that the angle which it 

makes with c remains always equal to {3. Before it reaches 
its final position b it must necessarily intersect a. In this way 
a triangle AB V C^ is determined, with the angles at A and B l 
respectively equal to a and {3. 

But, by WALLIS S hypothesis of the existence of similar 
figures, upon AB, the side homologous to AB^, we must be 
able to construct a triangle ABC similar to the triangle AB^ d. 
This is equivalent to saying that the straight lines a, b must 
meet in a point, namely, the third angular point of the triangle 
ABC. Therefore, etc. 

Wallis then seeks to justify the new position he has taken 
up. He points out that Euclid, in postulating the existence 
of a circle of given centre and given radius, [Post III.], practi 
cally admits the principle of similarity for circles. But even 
although intuition would support this view, the idea of form, 
independent of the dimensions of the figure, constitutes a 

Enclidis; disceptatio geometrica. Opera Math. t. II; p. 669 78 (Oxford, 
1693). This work by WALLIS contains two lectures given by him in 
the University of Oxford; the first in 1651, the second in 1663. It 
also contains the demonstration of NAsiR-EonlN. The part containing 
WALLIS S proof was translated into German by ENGEL and STACKEL in 
their Theorie der Parallellinien von Euclid bis auf Gauss, p. 21 36, 
(Leipzig, Teubner, 1895). We shall quote this work in future as 
Th. der P. 

Wallis s Proof. ij 

hypothesis, which is certainly not more evident than the Postu 
late of EUCLID. 

We remark, further, that WALLIS could more simply have 
assumed the existence of triangles with equal angles, or, as 
we shall see below, of only two unequal triangles whose 
angles are correspondingly equal. 

[cf. p. 29 Note i.] 

10 . The critical work of the preceding geometers is 
sufficient to show the historical development of our subject in 
the 1 6 th and i7 tn Centuries, so that it would be superfluous 
to speak of other able writers, such as, e. g., OLIVER of 
BURY [1604], LUCA VALERIO [1613], H. SAVILE [1621], 
A. TACQUET [1654], A. ARNAULD [I667]. 1 However, it seems 
necessary to say a few words on the question of the position 
which the different commentators on the Elements allot to 
the Euclidean hypothesis in the system of geometry. 

In the Latin edition of the Elements [1482], based upon 
the Arabian texts, by CAMPANUS [i3 th Century], this hypothesis 
finds a place among the postulates. The same may be said 
of the Latin translation of the Greek version by B. ZAMBERTI 
[1505], of the editions of LUCA PACIUOLO [1509], of N. TAR- 
TAGLIA [1543], of F. COMMANDING [1572], and of G. A.BoR- 
ELLI [1658]. 

On the other hand the first printed copy of the Ele 
ments in Greek, [Basle, 1533], contains the hypothesis among 
the axioms [Axiom XI]. In succession it is placed among the 
Axioms by F. CANDALLA [1556], C. S. CLAVIO [1574], GIOR 
DANO VITALE [1680], and also by GREGORY [1703], in his 
well-known Latin version of EUCLID S works. 

To attempt to form a correct judgment upon these dis- 

1 For fuller information on this subject cf. RICCARDI: Saggio 
di una bibliografia cuclidea. Mem. di Hologna, (5) T. I. p. 27 34, 


1 3 ! The Attempts to prove Euclid s Parallel Postulate. 

crepancies, due more to the manuscripts handed down from 
the Greeks than to the aforesaid authors, it will be an advan 
tage to know what meaning the former gave to the words 
postulates [aiTr||uaTa] and axioms [dHiujjuara]. 1 First of all 
we note that the word axioms is used here to denote what 
EUCLID in his text calls common notions [KOIVQI evvoiai]. 

PROCLUS gives three different ways of explaining the differ 
ence between the axioms and postulates. 

The first method takes us back to the difference between 
a problem and a theorem, A postulate differs from an axiom, 
as a problem differs from a theorem, says PROCLUS. By this we 
must understand that a postulate affirms the possibility of a 

The second method consists in saying that a postulate is 
a proposition with a geometrical meaning, while an axiom is a 
proposition common both to geometry and to arithmetic. 

Finally the third method of explaining the difference 
between the two words, given by PROCLUS, is supported by the 
authority of ARISTOTLE [384 322 B. C.]. The words axiom 
&b& postulate do not appear to be used by ARISTOTLE exclusive 
ly in the mathematical sense. An axiom is that which is true 
iji itself, that is, owing to the meaning of the words which it 
contains; a postulate is that which, although it is not an axiom, 
in the aforesaid sense, is admitted without demonstration. 

Thus the word axiom, as is more evident from an ex 
ample due to ARISTOTLE, \when equal things are subtracted from 
equal things the remainders are equal\ is used in a sense which 

i For the following, cf. PROCLUS, in the chapter entitled Pe 
tit a et axiomata. In a Paper read at the Third Mathematical Congress 
(Heidelberg, 1904) G. VAILATI has called the attention of students 
anew to the meaning of these words among the Greeks. Cf. : /- 
torno al significato Jella distinzione tra gli assiomi ed i postulate nella 
gcometria greca. Verh. des dritten Math. Kongresses, p. 575 581, 
(Leipzig, Teubner, 1905). 

Position of the Parallel Postulate. . \g 

corresponds, at any rate very closely, to that of the common 
notions of EUCLID, whilst the word postulate in ARISTOTLE has 
a different meaning from each of the two to which reference 
has just been made. 1 

Hence according as one or other of these distinctions be 
tween the words is adopted, a particular proposition would be 
placed among the postulates or among the axioms. If we 
adopt the first, only the first three of the five postulates of 
EUCLID, according to PROCLUS, have a right to this name, since 
only in these are we asked to carry out a construction [to 
join two points, to produce a straight line, to describe a circle 
whose centre and radius are arbitrary]. On the other hand, 
Postulate IV. [all right angles are equal], and Postulate V. ought 
to be placed among the axioms. 2 

1 Cf. ARISTOTLE: Analytica Posterior a. I. 10. $ 8. We quote in 
full this slightly obscure passage, where the philosopher speaks of 
the postulate: oaa ^ev ouv beiKTix ovra Xajapdvei auTO? nr] beiHac., 
TaOra lav |uev boKoOvra Xajupavrj TUJ jaavOdvovTi imoTiGeTai. Kai 
O*TIV oux duXux; UTr69eaic; dXXd Tcpo<; e^eivov j^6vov. Edv b f) 
|ur|be|uifi<; e*vouar|(; b6rjc f| Kai e vavTiat; vouar|<; XajufJavr], TO auto 
aireirai. Kai TOUTLU biaqpepei OuoOeaiq Kai airrnua, ?aTi YP 
aiiriiua T6 OirevavTiov ToO |uavedvovTO<; Tf) bor). 

2 It is right to remark that the Fifth Postulate can be enun 
ciated thus: 7^he common point of two straight lines can be found, when 
these two lines, cut by a transversal, form two interior angles on the 
same side whose sum is less than two right angles. Thus it follows 
that this postulate affirms, like the first three, the possibility of a 
construction. However this character disappears altogether, if it 
is enunciated, for example, thus: Through a point there passes only 
one parallel to a straight line ; or, thus : 7 ^wo straight lines which are 
parallel to a third line are parallel to each other. It would therefore 
appear that the distinction noted above is purely formal. However 
we must not let ourselves be deceived by appearances. The Fifth 
Postulate, in whatever way it is enunciated, practically allows the 
construction of the point of intersection of all the straight lines of 
a pencil with a given straight line in the plane of the pencil, one 
of these lines alone being excepted. It is true that there is a certain 


2O I- The Attempts to prove Euclid s Parallel Postulate. 

Again, if we accept the second or the third distinction, 
the five Euclidean postulates should all be included among 
the postulates. 

In this way the origin of the divergence between the var 
ious manuscripts is easily explained. To give greater weight 
to this explanation we might add the uncertainty which histor 
ians feel in attributing to EUCLID the postulates, common no 
tions and definitions of the First Book. So tar as regards the 
postulates, the gravest doubts are directed against the last 
two. The presence of the first three is sufficiently in accord 
with the whole plan of the work. 1 Admitting the hypothesis 
that the Fourth and Fifth Postulates are not Euclid s, even if 
it is against the authority of Geminus and Proclus, the ex 
treme rigour of the Elements would naturally lead the later 
geometers to seek in the body of the work all those pro 
positions which are admitted without demonstration. Now 
the one which concerns us is found stated very concisely in 
the demonstration of Bk. I. Prop. 29. From this, the sub 
stance of the Fifth Postulate could then be taken, and added 
to the postulates of construction, or to the axioms, according 
to the views held by the transcriber of EUCLID S work. 

Further, its natural place would be, and this is GREGORY S 
view, after Prop. 27, of which it enunciates the converse. 

Finally, we remark that, whatever be the manner of de 
ciding the verbal question here raised, the modern philo 
sophy of mathematics is inclined generally to suppress the 

difference between this postulate and the three postulates of con 
struction. In the latter the data are completely independent. In 
the former the data (the two straight lines cut by a transversal) are 
subject to a condition. So that the Euclidean Hypothesis belongs 
to a class intermediate between the postulates and axiom, rather 
than to the one or the other. 

i Cf. P. TANNERY: Sur Pauthenticitt dts axiomes (PEuclidc. Bull. 
d. Sc. Math. (2), T. VIII. p. 162175, (1884). 

Postulates and Axioms. 21 

distinction between postulate and axiom, which is adopted in 
the second and third of the above methods. The generally 
accepted view is to regard the fundamental propositions of 
geometry as hypotheses resting upon an empirical basis, 
while it is considered superfluous to place statements, which 
are simple consequences of the given definitions, among the 

Chapter II. 

The Forerunners of Non-Euclidean 

Gerolamo Saccheri [1667 1733]. 

ii. The greater part of the work of GEROLAMO SAC 
CHERI: Euclides ab omni naevo vindicates : sive conatus gco- 
nietricns quo stabiliuntur prima ipsa universae Geometriae 
Principia, [Milan, 1733], is devoted to the proof of the Fifth 
Postulate. The distinctive feature of SACCHERI S geometrical 
writings is to be found in his Logica detnotistrativa , [Turin, 
1697]. It is simply a particular method of reasoning, already 
used by EUCLID [Bk. IX. Prop. 1 2], according to which by 
assuming as hypothesis that the proposition which is to be proved 
is false, one is brought to the conclusion that it is true? 

Adopting this idea, the author takes as data the first 
twenty-six propositions of EUCLID, and he assumes as a hypo 
thesis that the Fifth Postulate is false. Among the consequences 
of this hypothesis he seeks for some proposition, which would 
entitle him to affirm the truth of the postulate itself. 

Before entering upon an exposition of SACCHERI S work, 
we note that EUCLID assumes implicitly that the straight line 
is infinite in the demonstration of Bk. I. 1 6 [the exterior angle 
of a triangle is greater than either of the interior and opposite 

1 Cf. G. VAILATI: Di nn* opera dimenticata del P. Gerolamo Sac- 
theri, Rivista Filosofica (1903). 

Saccheri s Quadrilateral. 23 

angles], since his argument is practically based upon the 
existence of a segment which is double a given segment. 

We shall deal later with the possibility of abandoning 
this hypothesis. At present we note that SACCHERI tacitly as 
sumes it, since in the course of his work he uses tibt proposition 
of the exterior angle. 

Finally, we note that he also employs the Postulate of 
Archimedes 1 - and the hypothesis of the continuity of the straight 
line, 2 to extend, to all the figures of a given type, certain pro 
positions admitted to be true only for a single figure of that 

12. The fundamental figure of SACCHERI is the two 
right-angled isosceles quadrilateral; that is, the quadrilateral of 
which two opposite sides are equal to each other and perpen 
dicular to the base. The properties of such a figure are de 
duced from the following Lemma I. , which can easily be 

If a quadrilateral ABCD has the consecutive angles A 
and B right angles, and the sides AD and BC equal, then the 
angle C is equal to the angle D [This is a special case of SAC 
CHERI S Prop. I.]; but if the sides AD and BC are unequal, of 
the two angles C, D, that one is greater which is adjacent to 
the shorter side, and vice versa. 

1 [The Postulate of Archimedes is stated by Hilbert thus: Let 
Ai be any point upon a straight line between the arbitrarily chosen 
points A and B. Take the points A 2 , A$, ... so that A-i lies 
between A and A 2 , A 2 between AI and A^ etc.; moreover let the 
segments AA*, A^A 2t A 2 A$, ... be all equal. Then among this 
series of points, there always exists a certain point A n , such that 
B lies between A and A n .} 

2 This hypothesis is used by SACCHERI in its intuitive form, 
viz.: a segment, which passes continuously from the length a to 
the length b t different from a, takes, during its variation, every 
length intermediate between a and b. 

24 II- The Forerunners of Non Euclidean Geometry. 

Let ABCD be a quadrilateral with t\vo right angles A 
and B, and two equal sides AD and BC (Fig. 9). On the 
Euclidean hypothesis the angles Cand D are also right angles. 
Thus, if we assume that they are able to be both obtuse, or 
both acute, we implicitly deny the Fifth Postulate. SACCHERI 
discusses these three hypotheses regarding the angles C, D. 
He named them: 

The Hypothesis of the Right Angle 

[<^ C= <^ D = i right angle] : 
The Hypothesis of the Obtuse Angle 

[< C = D > i right angle] : 
TJfo Hypothesis of the Acute A?igle 

[< C= <$:> < i right angle]. 
One of his first important results is the following: 
According as the Hypothesis of the Right Angle, of the 
Obtuse Angle, or of the Acute Angle is true in the two right- 
angled isosceles quadrilateral, we must have AB = CD, 
AB> CD, or AB < CD, respectively. [Prop. III.] 

In fact, on the Hypothesis of the Right Angle, by the 
preceding Lemma, we have immediately 

AB = CD. 

On the Hypothesis of the Obtuse Angle, the perpendicular 
OO at the middle point of the segment AB 
divides the fundamental quadrilateral into 
two equal quadrilaterals, with right angles at 
O and O . Since the angle D >> angle A, 
then we must have AO >> DO , by this 
Lemma. Thus AB> CD. 




On the Hypothesis of the Acute Angle these 
Flg 9- inequalities have their sense changed and 

we have 

AB < CD. 

Using the reductio ad absurdum argument, we obtain 
the converse of this theorem. [Prop. IV.] 

The Three Hypotheses. 25 

If the Hypothesis of the Right Angle is true in only one 
tase, then it is true in every other case. [Prop. V.] 

Suppose that in the two right-angled isosceles quadrilat 
eral ABCD the Hypothesis of the Right Angle is verified. 

In AD and BC (Fig. 10) take the points J7and K equi 
distant from AB; join ^ATand form the 
quadrilateral ABKH. 


If HK is perpendicular to AH and 
UK, the Hypothesis of the Right Angle is 
also verified in the new quadrilateral. 

If it is not, suppose that the angle 
AHK is acute. Then the adjacent angle A 
DHK is obtuse. Thus in the quadrilateral 
ABKH, from the Hypothesis of the Acute Angle, it follows 
that AB < HK\ while in the quadrilateral HKCD, from the 
Hypothesis of the Obtuse Angle, it follows that HK<^ CD. 

But these two inequalities are contradictory, since by 
}he Hypothesis of the Right Angle in the quadrilateral ABCD, 
AB = CD. 

Thus the angle AHK cannot be acute : and since by the 
same reasoning we could prove that the angle AHK cannot 
be obtuse, it follows that the Hypothesis of the Right Angle is 
also true in the quadrilateral ABKH. 

On AD and BC produced, take the points M, N equi 
distant from the base AB. Then the Hypothesis of the Right 
Angle is also true for the quadrilateral ABNM. In fact if 
AM\s a multiple of AD, the proposition is obvious. If AM 
is not a multiple of AD, we take a multiple of AD greater 
than AM [the Postulate of Archimedes], and from AD and 
BC produced cut off AP and BQ equal to this multiple. 
Since, as we have just seen, the Hypothesis of the Right Angle 
is true in the quadrilateral ABQP, the same hypothesis must 
also hold in the quadrilateral ABNM. 

Finally the said hypothesis must hold for a quadrilateral 


II. The Forerunners of Non-Euclidean Geometry. 

on any base, since, in Fig. i o, we can take as the base one 
of the sides perpendicular to AB. 

Note. This theorem of SACCHERI is practically contained 
in that of GIORDANO VITALE, stated on p. 1 5. In fact, refer 
ring to Fig. 7, the hypothesis 

is equivalent to the other 

<Z>= <J7=<:C=> i right angle. 
Hut from the former, there follows the equidistance of the 
two straight lines DC, AB*\ and thus the validity of the Hypo 
thesis of the Right Angle in all the two right-angled isosceles 
quadrilaterals, whose altitude is equal to the line DA, is 
established. The same hypothesis is also true in a quadri 
lateral of any height, since the line called at one time the 
base may later be regarded as the height. 

If the Hypothesis of the Obtuse Angle is true in only one 
case, then it is true in every other case. [Prop. VI.] 

Referring to the standard quadrilateral ABCD (Fig. 1 1), 
suppose that the angles C and D are ob 
tuse. Upon AD and BC take the points 
H and K equidistant from AB. 

In the first place we note that the 
segment HK. cannot be perpendicular to 
the two sides AD and BC, since in that 
case the Hypothesis of the Right Angle 
would be verified in the quadrilateral 
ABKH, and consequently in the fundamental quadrilateral. 
Let us suppose that the angle AHK is acute. Then 

1 It is true that GIORDANO in his argument refers to the points 
of the segment DC, which he shows are equidistant from the base 
AB of the quadrilateral. However the same argument is applicable 
to all the points which lie upon DC, or upon DC produced. Cf. 
BONOLA S Note referred to on p. 15. 










Proof for one Quadrilateral Sufficient. 27 

by the Hypothesis of the Acute Angle, HK >> AB. But as the 
Hypothesis of the Obtuse Angle holds in ABDC, we have 

AB> CD. 

Therefore HK> AB > CD. 

If we now move the straight line HK continuously, so that it 
remains perpendicular to the median OO of the fundamental 
quadrilateral, the segment HK, contained between the oppo 
site sides AD, BC, which in its initial position is greater than 
AB, will become less than AB in its final position DC. From 
the postulate of continuity we may then conclude that, 
between the initial position HK and the final position DC, 
there must exist an intermediate position H K , for which 
H K = AB. 

Consequently in the quadrilateral ABK H the Hypo 
thesis of the Right Angle would hold [Prop. III.J, and therefore, 
by the preceding theorem, the Hypothesis of the Obtuse Angle 
could not be true in ABCD. 

The argument is also valid if the segments AH, JBK arz 
greater than AD, since it is impossible that the angle AHK 
could be acute. Thus the Hypothesis of the Obtuse Angle holds 
in ABKH as well as in ABCD. 

Having proved the theorem for a quadrilateral whose 
sides are of any size, we proceed to prove it for one whose 
base is of any size: for example the base BK [cf. Fig. 12]. 

Since the angles K, PI, are obtuse, the 
perpendicular at K to KB will meet the 
segment AH in the point M, making the 

angle AMK obtuse [theorem of the ex 
terior angle]. 

Then in ABKM we have AB> KM, 
by Lemma I. Cut off from AB the segment A N 
BN equal to MK. Then we can construct 
the two right-angled isosceles quadrilateral BKMN, with the 
angle MNB obtuse, since it is an exterior angle of the triangle 

2$ II The Forerunners of Non-Euclidean Geometry. 

ANM. It follows that the Hypothesis of the Obtuse Angle 
holds in the new quadrilateral. 

Thus the theorem is completely demonstrated. 

If the Hypothesis of the Acute Angle is true in only one 
case, then it is true in every other case. [Prop. VII. ] 

This theorem can be easily proved by using the method 
of reductio ad absurdum. 

13. From the theorems of the last article SACCHER: 
easily obtains the following important result with regard to 

According as the Hypothesis of the Right Angle, the Hy 
pothesis of the Obtuse Angle, or the Hypothesis of the Acute 
Angle, is found to be true, the sum of the angles of a triangle 
tuill be respectively equil to, greater than, or less than two right 
angles. [Prop. IX.] 

Let ABC [Fig. 13] be a triangle of which B is a right 
D c angle. Complete the quadrilateral by draw 

ing AD perpendicular to AB and equal to 
BC; and jon CD. 

On the Hypothesis of the Right Angle, 
the two triangles ABC and ADC are equal. 
Therefore <)C BAG = < DC A. 
It follows immediately that in the tri 
angle ABC, 

+ < C = 2 right angles. 


Fig. 13. 

On the Hypothesis of the Obtuse Angle, 

since AB>DC, 
we have < ACS > <C DAC. 

1 This inequality is proved by SACCHERI in his Prop. VIII., 
and serves as Lemma to Prop. IX. It is, of course, Prop. 25 of 
EUCLID S First Book. 

The Sum of the Angles of a Triangle. 20 

Therefore, in this triangle we shall have 

*$: A + <^ B + < C> 2 right angles. 
<9# //fe Hypothesis of the Acute Angle, 

since AB<^DC, 
we have <: ^C^ < < ZX4C, 
and therefore, in the same triangle, 

<C A + ^B + ^C C< 2 right angles. 
The theorem just proved can be easily extended to the 
case of any triangle, by breaking the figure up into two right 
angled triangles. In Prop. XV. SACCHERI proves the converse, 
by a reductio ad absurdum. 

The following theorem is a simple deduction from these 
results : 

If the sum of the angles of a triangle is equal to, greater 
than, or less than two right angles in only one triangle, this 
sum will be respectively equal to, greater than, or less than frvo 
right angles in ei ery other triangle* 

This theorem, which SACCHERI does not enunciate ex 
plicitly, Legendre discovered anew and published, for the 
first and third hypotheses, about a century later. 

14. The preceding theorems on the two right- 
angled isosceles quadrilaterals were proved by SACCHERI, and 

1 Another of SACCHERI S propositions, which does not concern 
us directly, states that if the sum of the angles of only one quadri 
lateral is equal to, greater than, or less than four right angles, the 
Hypothesis of the Right Angle, the Hypothesis of the Obtuse Angle, or 
the Hypothesis of the Acute Angle would respectively be true. A note 
of SACCHERI S on the Postulate of WALLIS (cf. S 9} makes use of 
this proposition. He points out that WALLIS needed only to assume 
the existence of two triangles, whose angles were equal each to 
each and sides unequal, to deduce the existence of a quadrilateral 
in which the sum of the angles is equal to four right angles. From 
this the validity of the Hypothesis of the Right Angle would follow, 
and in its turn the Fifth Postulate. 

?O II. The Forerunners of Non-Euclidean Geometry. 

later by other geometers, with the help of the Postulate of 
Archimedes and the principle of continuity [cf. Prop. V., VI]. 
However DEHN* has shown that they are independent of 
these hypotheses. This can also be proved in an elementary 
way as follows. 2 

On the straight line r (Fig. 14) let two points B and D 
be chosen, and equal perpendiculars BA and DC be drawn 
to these lines. Let A and C be joined by the straight line s. 
The figure so obtained, in which evidently ^BAC= <^ DC A, 
is fundamental in our argument and we shall refer to it con 

Two points E, are now taken on s, of which the 
first is situated between A and C, and the second not. 

Further let the perpendiculars from E, to the line 
r meet it at F and F . 

The following theorems now hold: 

[If EF=AB,\ 
\. or |> , the angles BAG, DC A are right angles. 

[ E F =AB\ 

II. j or [ , the angles BAG, DC A are obtuse. 

I E F <AB] 

III. ] or }, the angles BAC, DCA are acute. 

I EF >AB\ 

We now prove Theorem I. [cf. Fig. 14.] 

From the hypothesis EF AB, the following equalities 
are deduced: 

1 Cf. Die Legendreschen Satze iiber die II inkelsiimme im Dreieck* 
Math. Ann. Bd. 53, p. 405 439 (1900). 

2 Cf. BONOLA, J teorcmi del Padre Gerolamo Saccheri sulla 
sow ma degli angoli di un triangolo e le ricerche di M. Dehn. Rend. 
Istituto Lombardo (2), Vol. XXXVIII. (1905). 

Postulate of Archimedes not needed. 

< BAE = ^: FEA, and <^ FEC - <c DCE. 
These, together with the fundamental equality 

are sufficient to establish the equality of the two angles FEA 
and FEC. 

E A E C 

F B F D 

Fig. 14. 

Since these are adjacent angles, they are both right 
angles, and consequently the angles BAG and DCA are 
right angles. 

The same argument is applicable in the hypothesis 
E F AB. 

We proceed to Theorem II [cf. Fig. 15]. 




~~~- - _ 


F B F D 

Fig. 15- 

Suppose, in the first place, EF ^> AB. From FE cut 
off FI= AB, and join / to A and C. 

Then the following equalities hold: 

< BAI= < F1A and DCJ = < FIC. 
Further, by the theorem of the exterior angle [Bk. I. 16], 
we have 

^2 U- Ih e Forerunners of Xon-Euclidean Geometry. 

4: FIA -f <: FIC> <C ,FW + < FEC = 2 right angles. 


<; .&4C + < Z><7/f > <3C ^7^ + ^ 7YC> 2 right angles. 

But, since < BAC= <^r Z?C4, 

it follows that ^BAC^> i right angle. . . . Q. E. D. 

In the second place, suppose that E F < AB. Then from 
F E produced cut off FT = BA, and join / to C and ^. 
The following relations, as usual, hold: 

^ F l A = ^C ^/ , ^F l C ^^ DCI ; 

$i I AE > <C / C 1 ^, <^ ^/ ^< < F l C. 
Combining these results, we deduce, first of all, that 

From this, if we subtract the terms of the inequality 

we obtain 

DCh = < BAC. 
But the two angles BAE and BAC are adjacent. Thus we 
have proved that ^C BAC\*> obtuse. Q. E. D. 

Theorem III. can be proved in exactly the same way. 
The converses of these theorems can now be easily 
shown to be true by the reductio ad absurdum method. In 
particular, if M and N are the middle points ot the two seg 
ments AC and BD, we have the following results for the 
segment MN which is perpendicular to both the lines AC 
and BD (Fig. 16). 

If <: BAC= -^ DC A = / right angle, then MN= AB. 
If < BAC= <:DCA > / right angle, then MN>AB. 
If < BAC= $: DC A < / right angle, then MN< AB. 
Further it is easy to see that 

(i) If < BAC = <_ DC A / right angle, 
then -}C FEM and i \ / E M are each i rig/it angle. 

Bonola s Proof. 


(ii) If^iBAC^ <C DC A > / r/^/ angle, 
then < FEM and <^ F E M are each obtuse. 

(iii) If <: BAG = ^ Z>6W < / right angle, 
then < FE M and < F E M are each acute. 

E \ E 

M C 

F B F 

N D 

Fig. 1 6. 

In fact, in Case (i), since the lines r and s are equi 
distant, the following equalities hold: 
^NMA = ^FEM= <: BAC= ^F E M=i rightangle. 

To prove Cases (iij and (iii), it is sufficient to use the 
reductio ad absurdum method, and to take account of the 
results obtained above. 

Now let P be a point on the line MN, not contained 
between J/and j^V(Fig. 1 7). Let RP be the perpendicular to 
MWaxid ^?A"the perpendicular to BD. This last perpend 
icular will meet AC in a point H. On this understanding 
the preceding theorems immediately establish the truth of 
the following results: 

If < BAM = i right angle, then < KHM and < KRP 
are each equal to / right angle. 

If ^ BAM> T right angle, then <^ KHM and < KRP 
are each greater than i right angle. 

If < BAM < i right angle, then < KHM and <^ KRP 
are each less than i right angle. 

These results are also true, as can easily be seen, if the 
point P falls between M and N. 

In conclusion, the last three theorems, which clearly 


II. The Forerunners of Non-Euclidean Geometry. 

coincide with Saccheri s theorems upon the two right-angled 
isosceles quadrilateral, are equivalent to the following result, 
proved without using Archimedes Postulate: 





M C 

K B 

N D 

Fig. 17. 

If the truth of the Hypothesis of the Right Angle, of the 
Obtuse Angle, or of the Acute Angle, respectively, is known in 
only one case, its truth is also known in every other case. 

If we wish now to pass from the theorems on quad 
rilaterals to the corresponding theorems on triangles, we need 
only refer to SACCHERI S demonstration [cf. p. 28], since this 
part of his argument does not in any way depend upon 
the postulate in question. 

We have thus obtained the result which was to be 

15. To make our exposition of SACCHERI S work 
more concise, we take from Prop. XL and XII. the contents 
of the following Lemma II: 

Let ABC be a triangle of which C is a right angle: let 
H be the middle point of AB, and K the foot of the perpen 
dicular from H upon AC. Then we shall have 

AK = KC, on the Hypothesis of the Right Angle; 
KC, on the Hypothesis of the Obtuse Angle; 
KC, on the Hypothesis of the Acute Angle. 

On the Hypothesis of the Right Angle the result is 

The Projection of a Line. 


On the Hypothesis of the Obtuse Angle, since the sum of 
the angles of a quadrilateral is greater than four right angles, 
it follows that ^AHK<_^HBC. Let HL be the perpendi 
cular from H to BC (Fig. 18). Then the result just obtained, 
and the fact that the two triangles AHK, HBL have equal 
hypotenuses, give rise to the folio wing inequality: AK<^HL. 
But the quadrilateral HKCL has three right angles and there 
fore the angle H is obtuse {Hypothesis of the Obtuse Angle]. 

It follows that 

HL < KC, 

and thus 

The third part of this Lemma can be proved in the 
same way. 

It is easy to extend this Lemma as follows (Fig. 1 9) : 

A A 

Fig. 19. 

Lemma III. If on the one arm of an angle A equal seg 
ments AAi, A-iA 2) A 2 A Z) . . . are taken, and AA^ , A^A 2 \ 
A 2 A$. . . are their projections upon the other arm of the angle, 
then the following results are true: 

on the Hypothesis of the Right Angle; 
on the Hypothesis of the Obtuse Angle; 

on the Hypothesis of the Acute Angle. 

To save space the simple demonstration is omitted. 


The Forerunners of Non-Euclidean Geometry. 

We can now proceed to the proof of Prop. XI. and XII. 
of Saccheri s work, combining them in the following theorem : 

On the Hypothesis of the Right Angle and on the Hypo 
thesis of the Obtuse Angle, a line perpendicular to a given 
straight line and a line cutting it at an acute angle intersect 
each other. 

Fig. ao. 

Let (Fig. 20) LP and AD be two straight lines of which 
the one is perpendicular to AP, and the other is inclined to 
AP at an acute angle DAP. 

After cutting off in succession equal segments AD, DF^, 
upon AD, draw the perpendiculars DB and F^M^ upon the 
line AP. 

From Lemma III. above, we have 

BMi 5 AB, 

or AMi |> 2 AB } 

on the two hypotheses. 

Now cut off Fj_F 2 equal to AF IJ from AF^ produced, 
and let M 2 be the foot of the perpendicular from F 2 upon AP. 
Then we have 

AM 2 <> 2 AM i, 
and thus 

AM 2 > 2> AB. 

This process can be repeated as often as we please. 

In this way we would obtain a point F H upon the line 
AD such that its projection upon the line AP would deter 
mine a segment AM* satisfying the relation 

Two Hypotheses give Postulate V. 



But if n is taken sufficiently great, [by the Postulate of 
Archimedes^ we would have 

and therefore 

AM n > AP. 

Therefore the point P lies upon the side AM n of the right- 
angled triangle AM H F n . The perpendicular PL cannot 
intersect the other side of this triangle; therefore it cuts the 
hypotenuse. 2 Q. E. D. 

It is now possible to prove the following theorem : 

The Fifth Postulate is true on the Hypothesis of the 
Right Angle and on the Hypothesis of the Obtuse Angle [Prop. 

Let (Fig. 21) AB, CD be two straight lines cut by the 
line AC. 

Let us suppose that 

< BAG + ACD < 2 right angles. 

Then one of the angles 
BAG, ACD, for example the 
first, will be acute. 

From C draw the perpen 
dicular CH upon AB. In the 
triangle ACH, from the hypo 
theses which have been made, 
we shall have 

^A+ < C+ 4 


Fig. 21. 

H> 2 right angles. 

1 The Postulate of Archimedes, of which use is here made, 
includes implicitly the infinity of the straight line. 

2 The method followed by SACCHERI in proving this theorem 
is practically the same as that of NASIR-EDDIN. However Nastr- 
EDDIN only deals with the Hypothesis of the Right Angle, as he had 
formerly shown that the sum of the angles of a triangle is equal 
to two right angles. It is right to remember that SACCHERI was 
familiar with and had criticised the work of the Arabian Geometer. 

38 H- The Forerunners of Non-Euclidean Geometry. 

But we have assumed that 

< BAG + <^ ACD < 2 right angles. 
These two results show that 

<: AHC > <; HCD. 

Thus the angle HCD must be acute, as J? is a right angle. 
It follows from Prop. XL, XII. that the lines AB and CD 
intersect. 1 

This result allows SACCHERI to conclude that the Hypo 
thesis of the Obtuse Angle is false [Prop. XIV.]. In fact, on 
this hypothesis EUCLID S Postulate holds [Prop. XIII. J, and 
consequently, the usual theorems which are deduced from 
this postulate also hold. Thus the sum of the angles of the 
fundamental quadrilateral is equal to four right angles, so 
that the Hypothesis of the Right Angle is true. 2 

16. But SACCHERI wishes to prove that the Fifth 
Postulate is true in every case. He thus sets himself to 
destroy the Hypothesis of the Acute Angle. 

To begin with he shows that on this hypothesis, a straight 
line being given, there can be drawn a perpendicular to it and 
a line cutting it at an acute angle, which do not intersect each 
other [Prop. XVII.]. 

To construct these lines, let ABC (Fig. 22) be a triangle 
of which the angle C is a right angle. At B draw BD mak 
ing the angle ABD equal to the angle BAC. Then, on the 

1 This proof is also found in the work of NASIR-DDIN, which 
evidently inspired the investigations of SACCHERI. 

2 It should be noted that in this demonstration SACCHERI 
makes use of the special type of argument of which we spoke in 
S II. In fact, from the assumption that the Hypothesis of the Ob 
tuse Angle is true, we arrive at the conclusion that the Hypothesis 
of the Right Angle is true. This is a characteristic form taken in 
such cases by the ordinary reductio ad absurdum argument. 

Saccheri and the Third Hypothesis. 


Hypothesis of the Acute Angle, the angle CBD is acute, and 
of the two lines CA, BD, which do not meet [Bk. I. 27], 
one makes a right angle with BC. 

In what follows we consider only the Hypothesis of the 
Acute Angle. 

Let (Fig. 23) a, b be two straight lines in the same plane 
which do not meet. 


B B 

Kis. 22. Fig. 23. 

From the points A lt A 2 , on a draw perpendiculars 
AT.BI, A 2 B 2 to b. 

The angles A lt A 2 of the quadrilateral thus obtained 
can be 

(i) one right, and one acute: 

(ii) both acute: 

(in) one acute and one obtuse. 

In the first case, there exists already a common per 
pendicular to the two lines a, b. 

In the second case, we can prove the existence of such 
a common perpendicular by using the idea of continuity 
[SACCHERI, Prop. XXII.]. In fact, if the straight line A^ B^ is 
moved continuously, while kept perpendicular to b, until it 
reaches the position A 2 B 2) the angle B 1 A 1 A 2 starts as an 
acute angle and increases until it becomes an obtuse angle. 
There must be an intermediate position AB in which the 
angle BAA 2 is a right angle. Then AB is the common 
perpendicular to the two lines a, b. 

In the third case, the lines a, b do not have a common 


II. The Forerunners of Non-Euclidean Geometry. 

perpendicular, or, if such exists, it does not fall between B^ 
and B 2 . 

Evidently there will be no such perpendicular if, for all 
the points A r situated upon a, and on the same side of A lt 
the quadrilateral B^A^A r B r has always an obtuse angle a.tA r . 

With this hypothesis of the existence of two coplanar 
straight lines which do not intersect, and have no common 
perpendicular, SACCHERI proves that such lines always ap 
proach nearer and nearer to each other [Prop. XXIII.], and that 
their distance apart finally becomes smaller than any segment, 
taken as small as we please [Prop. XXV.]. In other words, 
if there are two coplanar straight lines, which do not cut 
each other, and have no common perpendicular, then these 
lines must be asymptotic to each other. 1 

To prove that such asymptotic lines effectively exist, 
SACCHERI proceeds as follows: 2 

Fig. 24. 

Among the lines of the pencil through A, coplanar with 
the line , there exist lines which cut b, as, e. g., the line 
AB perpendicular to b\ and lines which have a common 

1 With this result the question raised by the Greeks, as to 
the possibility of asymptotic lines in the same plane, is answered 
in the affirmative. Cf. p. 3. 

2 The statement of SACCHERI S argument upon the asymptotic 
lines differs in this edition from that given in the Italian and 
German editions. The changes introduced were suggested to me 
by some remarks of Professor CARSLAW. 

The Existence of Asymptotic Lines A\ 

perpendicular with b, as, e. g., the line A A perpendicular 
to AB [cf. Fig. 24]. 

If AP cuts b, every other line of the pencil, which 
makes a smaller angle with AB than the acute angle BAP, 
also cuts b. On the other hand, if the line A Q, different from 
AA } has a common perpendicular with b, every other line, 
which makes with AB a larger acute angle than the angle 
BAQ, has a common perpendicular with b [cf. 39, 
case (ii).] 

Also it is clear that, if we take the lines of the pencil 
through A, from the ray AB towards the ray AA , we shall 
not find, among those which cut b, any line which is the last 
line of that set. In other words, the angles BAP, which the 
lines AP, cutting b, make with AB, have an upper limit, the 
angle BAX, such that the line AX does not cut b. 

Then SACCHERI proves [Prop. XXX.] that, if we start with 
AA and proceed in the pencil through A in the direction 
opposite to that just taken, we shall not find any last line in 
the set of lines which have a common perpendicular with b\ 
that is to say, the angles BA Q, where A Q has a common 
perpendicular with b, have a lower limit, the angle BA Y, 
such that the line A Y does not cut b and has not a com 
mon perpendicular with b. 

It follows that A Y is a line asymptotic to b. 

Further SACCHERI proves that the two lines AX and A Y 
coincide [Prop. XXXII.]. His argument depends upon the 
consideration of points at infinity; and it is better to sub 
stitute for it another, founded on his Prop. XXL, viz., On the 
Hypothesis of the Right Angle, and on that of the Acute Angle, 
the distance of a point on one of the lines containing an angle 
from the other bounding line increases indefinitely as this point 
moves further and further along the line. 


II. The Forerunners of Non-Euclidean Geometry. 
The suggested argument is as follows: 

Fig. 25. 

If AX [Fig. 25] does not coincide with A Y } we can take 
a point P on A Y, such that the perpendicular PP r from P 
to AX satisfies the inequality 

( 1 1 Pf> AB. [Prop. XXL] 

On the other hand, if PQ is the perpendicular from P to , 
the property of asymptotic lines [Prop. XXIII] shows that 

But P is on the opposite side of AX from b. 
therefore PQ > PP. 

Combining this inequality with the preceding, we find that 

which contradicts (i). 

Hence AX coincides with A Y. 

We may sum up the preceding results in the following 
theorem : 

On the Hypothesis of the Acute Angle, there exist in the 
pencil of lines through A two lines p and q, asymptotic to b, 
one towards the right, and the other towards the left, which 
divide the pencil into two parts. The first of these consists of 
the lines which intersect b, and the second of those which have a 
common perpendicular with /V. 1 

1 In SACCHERI S work there will be found many other inter 
esting theorems before he reaches this result. Of these the 

Saccheri s Conclusion. 


17. At this point SACCHERI attempts to come to a 
decision, trusting to intuition and to faith in the validity of 
the Fifth Postulate rather than to logic. To prove that the 
Hypothesis of the Acute Angle is absolutely false, because it is 
repugnant to the nature of the straight line [Prop. XXXIIL] he 
relies upon five Lemmas, spread over sixteen pages. In sub 
stance, however, his argument amounts to the statement 
that if the Hypothesis of the Acute Angle were true, the 
lines p (Fig. 26) and b would have a common perpendicular 
at their common point at inft?iity, which is contrary to the 
nature of the straight line. The so-called demonstration of 
SACCHERI is thus founded upon the extension to infinity of 
certain properties which are valid for figures at a finite 

However, SACCHERI is not satisfied with his reasoning 
and attempts to reach the wished-for proof by adopting 
anew the old idea of equidistance. It is not worth while to 
reproduce this second treatment as it does not contain any 
thing of greater value than the discussions of his prede 

Still, though it failed in its aim, SACCHERI S work is of 
great importance. In it the most determined effort had been 
made on behalf of the Fifth Postulate; and the fact that he 
did not succeed in discovering any contradictions among 
the consequences of the Hypothesis of the Acute Angle, could 
not help suggesting the question, whether a consistent log 
ical geometrical system could not be built upon this hypo- 

following is noteworthy: If two straight lines continually approach 
each other and their distance apart remains always greater than a 
given segment, then the Hypothesis of the Acute Angle is impossible. 
Thus it follows that, if we postulate the absence of asymptotic 
straight lines, we must accept the truth of the Euclidean hypo 

A A H. The Forerunners of Non-Euclidean Geometry. 

thesis, and the Euclidean Postulate be impossible of demon 
stration. 1 

Johann Heinrich Lambert [1728 1777]- 
18. It is difficult to say what influence SACCHERI S 
work exercised upon the geometers of the i8 th century. 
However, it is probable that the Swiss mathematician 
LAMBERT was familiar with it, 2 since in his Theorie der Par- 
allellinien [1766] he quotes a dissertation by G. S. KLUGEL 
[ J 739 i8i2p, where the work of the Italian geometer 
is carefully analysed. LAMBERT S Theorie der Parallellinien 
was published after the author s death, being edited by 
J. BERNOULLI and C. F. HINDENBURG. It is divided into 
three parts. The first part is of a critical and philosophical 
nature. It deals with the two-fold question arising out of the 
Fifth Postulate : whether it can be proved with the aid of 
the preceding propositions only, or whether the help of some 
other hypothesis is required. The second part is devoted to 

1 The publication of SACCHERI S work attracted considerable 
attention. Mention is made of it in two Histories of Mathematics: 
that of J. C. HEILBRONNER (Leipzig, 1742) and that of MONTUCLA 
(Paris, 1758). Further it is carefully examined by G. S. KLUGEL 
in his dissertation noted below (Note (3)). Nevertheless it was 
soon forgotten. Not till 1889 did E. BELTRAMI direct the attention 
of geometers to it again in his Note: Un precursor e italiano 
di Legendre e di Lobatschewsky. Rend. Ace. Lincei (4), T. V. p. 441 
448. Thereafter SACCHERI S work was translated into English by 
G. B. HALSTED (Amer. Math. Monthly, Vol. I. 1894 et seq.); into 
German, by ENGEL and STCKEL (Th. der P. 1895); into Italian, 
by G. Boccardini (Milan, Hoepli, 1904). 

2 Cf. SEGRE: Congetture intorno alia influenza di Girolamo 
Saccheri sulla formazione della geometria non euclidea. Atti Ace. 
Scienze di Torino, T. XXXVIII. (1903). 

3 Conatuum praecipuorum theoriam parallelarum demonstrandi 
recensio, quam publico examini submittent A. G. Kaestner et auctor 
respondens G. S. Kliigel, (Gottingen, 1763). 

Lambert s Three Hypotheses. 


the discussion of different attempts in which the Euclidean 
Postulate is reduced to very simple propositions, which 
however, in their turn, require to be proved. The third, and 
most important, part contains an investigation resembling 
that of SACCHERI, of which we now give a short summary. 1 

19. LAMBERT S fundamental figure is a quadrilateral 
with three right angles, and three hypotheses are made as to 
the nature of the fourth angle. The first is the Hypothesis 
of the Right Angle] the second, the Hypothesis of the Obtuse 
Angle; and the third, the Hypothesis of the Acute Angle. Also 
in his treatment of these hypotheses the author does not 
depart far from SACCHERI S method. 

Theyfr-r/ hypothesis leads easily to the Euclidean system. 

In rejecting the second hypothesis, LAMBERT relies upon 
a figure formed by two straight lines a, b, perpendicular to 
a third line AB (Fig. 27). From points B, B^, B 2 >..B n , 
taken in succession upon B Bj B 2 8 B 

the line b, the perpen 
diculars, BA, t A lt B 2 A 2 , 
. . B n A, t are drawn to the 
line a. He proves, in the 
first place, that these per 
pendiculars continually 
diminish, starting from the perpendicular BA. Next, that 
the difference between each and the one which succeeds it 
continually increases. 

Therefore we have 

BAB H A n > n (BAB^A^. 

But, if n is taken sufficiently large, the second member 

Fig. 27- 

1 Cf. Magazin fur reine und angewandte Math., 2. Stuck, 
p. 137164. 3. Stuck, p. 325358, (1786). LAMBERT S work was 
again published by ENGEL and STACKEL (Tk. der P.] p. 135 208, 
preceded by historical notes on the author. 

A& II. The Forerunners of Non-Euclidean Geometry. 

of this inequality becomes as great as we please [Postulate 
of Archimedes]*, whilst the first member is always less than 
BA. This contradiction allows LAMBERT to declare that the 
second hypothesis is false. 

In examining the third hypothesis, LAMBERT again avails 
himself of the preceding figure. He proves that the perpen 
diculars BA, B^Ai, . . B n A H continually increase, and that 
at the same time the difference between each and the one 
which precedes it continually increases. As this result does 
not lead to contradictions, like SACCHERI he is compelled to 
carry his argument further. Then he finds, that, on the third 
hypothesis the sum of the angles of a triangle is less than 
two right angles; and going a step further than SACCHERI, 
he discovers that the defect of a polygon, that is, the differ 
ence between 2 (;/ 2) right angles and the sum of its angles, 
is proportional to the area of the polygon. This result can 
be obtained more easily by observing that both the area and 
the defect of a polygon, which is the sum of several others, 
are, respectively, the sum of the areas and of the defects of 
the polygons of which it is composed. 2 

20. Another remarkable discovery made by LAMBERT 
has reference to the measurement of geometrical magnitudes. 
It consists precisely in this, that, whilst in the ordinary geo 
metry only a relative meaning attaches to the choice of a 

1 The Postulate of Archimedes is again used here in a form 
which assumes the infinity of the straight line (cf. SACCHERI, Note 
P- 37). 

2 It is right to point out that in the Hypothesis af the Acute 
Angle SACCHERI had already met the defect here referred to, and 
also noted implicitly that a quadrilateral, made up of several 
others, has for its defect the sum of the defects of its parts (Prop. 
XXV). However he did not draw any conclusion from this as to 
the area being proportional to the defect. 

Relative and Absolute Units. A7 

particular unit in the measurement of lines, in the geometry 
founded upon the third hypothesis, we can attach to it an 
absolute meaning. 

First of all we must explain the distinction, which is 
here introduced, between absolute and relative. In many 
questions it happens that the elements, supposed given, can 
be divided into two groups, so that those of faz first group 
remain fixed, right through the argument, while those of the 
second groiip may vary in a number of possible cases. When 
this happens, the explicit reference to the data of the first 
group is often omitted. All that depends upon the varying 
data is considered relative; all that depends upon the fixed 
data is absolute. 

For example, in the theory of the Domain of Ration 
ality, the data of the second group [the variable data] are 
taken as certain simple irrationalities [constituting a base\, 
and the first group consists simply of unity [i], which is 
often passed over in silence as it is common to all domains. 
In speaking of a number, we say that it is rational relatively 
to a given base, if it belongs to the domain of rationality 
defined by that base. We say that it is rational absolutely, 
if it is proved to be rational with respect to the base i, 
which is common to all domains. 

Passing to Geometry, we observe that in every actual 
problem, we generally take certain figures as given and 
therefore the magnitudes of their parts. In addition to these 
variable data [of the second group\, which can be chosen in 
an arbitrary manner, there is always implicitly assumed the 
presence of the fundamental figures, straight lines, planes, 
pencils, etc. [fixed data or of the first group}. Thus, every 
construction, every measurement, every property of any 
figure ought to be held as relative, if it is essentially relative 
to the variable data. It ought, on the other hand, to be 
spoken of as absolute, if it is relative only to the fixed data 

^8 II. The Forerunners of Non-Euclidean Geometry. 

[the fundamental figures], or, if, being enunciated in terms 
of the variable data, it only appears to depend upon them, 
so that it remains fixed when these vary. 

In this sense it is clear that in ordinary geometry the 
measurement of lines has necessarily a relative meaning. 
Indeed the existence of similar figures does not allow us in 
any way to individualize the size of a line in terms of funda 
mental figures [straight line, pencil, etc.]. 

For an angle on the other hand, we can choose a method 
of measurement which expresses one of its absolute pro 
perties. It is sufficient to take its ratio to the angle of a 
complete revolution, that is, to the entire pencil, this being 
one of the fundamental figures. 

We return now to LAMBERT and his geometry corre 
sponding to the third hypothesis. He observed that with 
every segment we can associate a definite angle, which can 
easily be constructed. From this it follows that every seg 
ment is brought into correspondence with the fundamental 
figure [the pencil]. Therefore, in the new [hypothetical] 
geometry, we are entitled to ascribe an absolute meaning 
also to the measurement of segments. 

To show in the simplest way how to every segment we 
can find a corresponding angle, and thus obtain an ab 
solute numerical measurement of lines, let us imagine an 
equilateral triangle constructed upon every segment. We 
are able to associate with every segment the angle of the 
triangle corresponding to it and then the measure of this 
angle. Thus there exists a one-one correspondence between 
segments and the angles comprised between certain limits. 

But the numerical representation of segments thus ob 
tained does not enjoy the distributive property which belongs 
to lengths. On taking the sum of two segments, we do not 
obtain the sum of the corresponding angles. However, a 
function of the angle, possessing this property, can be ob- 

The Absolute Unit of Length. AQ 

tained, and we can associate with the segment, not the said 
angle, but this function of the angle. For every value of the 
angle between certain limits, such a function gives an absolute 
measure of segments. The absolute unit of length is that 
segment for which this function takes the value i. 

Now if a certain function of the angle is distributive in 
the sense just indicated, the product of this function and an 
arbitrary constant also possesses that property. It is there 
fore clear that we can always choose this constant so that 
the absolute unit segment shall be that segment which corre 
sponds to any assigned angle: e. g., 45. The possibility of 
constructing the absolute unit segment, given the angle, de 
pends upon the solution of the following problem: 

To construct, on the Hypothesis of the Acute Angle, an 
equilateral triangle with a given defect. 

So far as regards the absolute measure of the areas of 
polygons, we remark that it is given at once by the defect 
of the polygons. We can also assign an absolute measure 
for polyhedrons. 

But with our intuition of space the absolute measure 
of all these geometrical magnitudes seems to us impossible. 
Hence if we deny the existence of an absolute unit for segments, 
we can, with Lambert, reject the third hypothesis. 

21. As LAMBERT realized the arbitrary nature of this 
statement, let it not be supposed that he believed that he 
had in this way proved the Fifth Postulate. 

To obtain the wished-for proof, he proceeds with his 
investigation of the consequences of the third hypothesis, but 
he only succeeds in transforming his question into others 
equally difficult to answer. 

Other very interesting points are contained in the 
Theorie der Parallellinien, for example, the close resemblance 


CQ n. The Forerunners of Non-Euclidean Geometry. 

to spherical geometry 1 of the plane geometry which would 
hold, if the second hypothesis were valid, and the remark that 
spherical geometry is independent of the Parallel Postulate. 
Further, referring to the third hypothesis, he made the follow 
ing acute and original observation: From this I should al 
most conclude that the third hypothesis would occur in the case 
of an imaginary sphere. 

He was perhaps brought to this way of looking at the 
question by the formula (A -\-B-\- C TT) r 2 , which expresses 
the area of a spherical triangle. If in this we write for the 
radius r, the imaginary radius V-I r we obtain 

r 2 [TT A B C); 

that is, the formula for the area of a plane triangle on 
LAMBERT S third hypothesis. 2 

22. LAMBERT thus left the question in suspense. In 
deed the fact that he did not publish his investigation allows 
us to conjecture that he may have discovered another way 
of regarding the subject. 

Further, [it should be remarked that, from the general 
want of success of these attempts, the conviction began to 
be formed jn the second half of the >i8 th Century that it 
would be necessary to admit the Euclidean Postulate, or 
some other equivalent postulate, without proof. 

In Germany, where the writings upon the question 
followed closely upon each other, this conviction had al 
ready assumed a fairly definite form. We recognize it in 
A. G. KXSTNER, 3 a well-known student of the theory of 
parallels, and in his pupil, G. S. KLUGEL, author of the 

1 In fact, in Spherical Geometry the sum of the angles of a 
quadrilateral is greater than four right angles, etc. 

2 Cf. ENGEL u. STACKEL; Th. der P. p. 146. 

3 For some information about KASTNER, cf. ENGEL u. STACKEL; 
Th. der P. p. 139141. 

Kliigel s Work. i? I 

valuable criticism of the most celebrated attempts to de 
monstrate the Fifth Postulate, referred to on p. 44 [note 3]. 
In this work KLUGEL finds each of the proposed proofs 
insufficient and suggests the possibility of non-intersecting 
straight lines being divergent \Moglich ware es freilich, daft 
Gerade, die sich nihct schneiden, voneinander abweichen\ He 
adds that the apparent contradiction which this presents is 
not the result of a rigorous proof, nor a consequence of the 
definitions of straight lines and curves, but rather something 
derived from experience and the judgment of our senses. 
[Dafi so etwas widersinnig ist, wissen wir nicht infolge strenger 
Schlusse oder vermoge deutlicher Begriffe von der geraden und 
der krummen Linie, vielmehr durch die Erfahrung und durch 
das Urteil unserer Augen\. 

The investigations of SACCHERI and LAMBERT tend to 
confirm KLUGEL S opinion, but they cannot be held to be 
a proof of the impossibility of demonstrating the Euclidean 
hypothesis. Neither would a proof be reached if we proceed 
ed along the way opened by these two geometers, and de 
duced any number of other propositions, not contradicting 
the fundamendal theorems of geometry. 

Nevertheless that one should go forward on this path, 
without SACCHERI S presupposition that contradictions would 
be found there, constitutes historically the decisive step in the 
discovery that EUCUD S Postulate could not be proved, and 
in the creation of the Non-Euclidean geometries. 

But from the work of SACCHERI and LAMBERT to that of 
LOBATSCHEWSKY and BOLYAI, which is based upon the above 
idea, more than half a century had still to pass ! 

The French Geometers towards the End of the 

i8*h Century. 

23. The critical study of the theory of parallels, 
which had already led to results of great interest in Italy and 


C2 II. The Forerunners of Non-Euclidean Geometry. 

Germany, also made a remarkable advance in France to 
wards the end of the 18^ Century and the beginning of 
the 19 th . 

D ALEMBERT [1717 1783] , in one of his articles on 
geometry, states that La definition et les proprieties de la 
ligne droite, ainsi que des lignes paralleles sont 1 ecueil et 
pour ainsi dire le scandale des elements de Geometric. l 
He holds that with a good definition of the straight line 
both difficulties ought to be avoided. He proposes to define 
a parallel to a given straight line as any other coplanar 
straight line ; which joins two points which are on the same 
side of and equally distant from the given line. This definition 
allows parallel lines to be constructed immediately. However 
it would still be necessary to show that these parallels are 
equidistant. This theorem was offered, almost as a challenge, 
by D ALEMBERT to his contemporaries. 

24. DE MORGAN, in his Budget of Paradoxes 2 , relates 
that LAGRANGE [1736 1813], towards the end of his life, 
wrote a memoir on parallels. Having presented it to the 
French Academy, he broke off his reading of it with the ex 
clamation: II faut que j y songe encore! and he withdrew 
the MSS. 

Further HOUEL states that LAGRANGE, in conversation 
with BIOT, affirmed the independence of Spherical Trigon 
ometry from EUCLID S Postulate. 3 In confirmation of this 
statement it should be added that LAGRANGE had made a spe 
cial study of Spherical Trigonometry, 4 and that he inspired, 

1 Cf. D ALEMBERT: Melanges de Litterature, d Histoire, tt 
de Philosophic, T. V. S II ( I 759) Also: Encyclopedic Methodique 
Mathematique ; T. II. p. 519, Article: Paralleles (1785). 

2 A. DE MORGAN: A Budget of Paradoxes, p. 173. (London, 1872). 

3 Cf. J. HoCzL: Essai critique sur les principes fondamentaux 
de la geometric elementaire, p. 84, Note (Paris, G. VlLLARS, 1883). 

4 Cf. Miscellanea Taurinensia, T. II. p. 299322 (176061). 

D Alembert, Lagrange, and Laplace. 53 

if he did not write, a memoir l Sur les principes fondamentaux 
de la Mecanique [1760 i] 1 , in which FONCENEX discussed 
a question of independence, analogous to that above noted 
for Spherical Trigonometry. In fact, FONCENEX shows that 
the analytical law of the Composition of Forces acting at a 
point does not depend on the Fifth Postulate, nor upon any 
other which is equivalent to it. 2 

25. The principle of similarity, as a fundamental 
notion, had been already employed by WALLIS in 1663 [cf. 
S 9]. It reappears at the beginning of the 19 th Century, sup 
ported by the authority of two famous geometers: L. N. M. 
CARNOT [1753 1823] and LAPLACE [1749 1827]. 

In a Note [p. 481] to his Geometric de Position [1803] 
CARNOT affirms that the theory of parallels is allied to the 
principle of similarity, the evidence for which is almost on 
the same plane as that for equality, and that, if this idea is 
once admitted, it is easy to establish the said theory rigorously. 

LAPLACE [1824] observes that NEWTON S Law [the Law 
of Gravitation], by its simplicity, by its generality and by the 
confirmation which it finds in the phenomena of nature, must 
be regarded as rigorous. He then points out that one of its 
most remarkable properties is that, if the dimensions of all 
the bodies of the universe, their distances from each other, 
and their velocities, were to decrease proportionally, the 
heavenly bodies would describe curves exactly similar to 
those which they now describe, so that the universe, reduced 
step by step to the smallest imaginable space, would always 
present the same phenomena to its observers. These pheno 
mena, he continues, are independent of the dimensions of the 
universe, so that the simplicity of the laws of nature only allows 
the observer to recognise their ratios. Referring again to this 

* Cf. LAGRANGE: Oeuvres, T. VII. p. 331 363. 
2 Cf. Chapter VI. 

4 II- The Forerunners of Non-Euclidean Geometry. 

astronomical conception of space, he adds in a Note: The 
attempts of geometers to prove EUCLID S Postulate on Parallels 
have been up till now futile. However no one can doubt this 
postulate and the theorems which EUCLID deduced from it. Thus 
the notion of space includes a special property, self-evident, 
without which the properties of parallels cannot be rigorously 
established. The idea of a bounded region, e. g., the circle, 
contains nothing which depends on its absolute magnitude. 
But if we imagine its radius to diminish, we are brought 
without fail to the diminution in the same ratio of its circum 
ference and the sides of all the inscribed figures. This pro 
portionality appears to me a more natural postulate than 
that of EUCLID, and it is worthy of note that it is discovered 
afresh in the results of the theory of universal gravitation. l 

26. Along with the preceding geometers, it is right 
also to mention J. B. FOURIER [1768 1830], for a discussion 
on the straight line which he carried on with MONGE. Z To 
bring this discussion into line with the investigations on 
parallels, we need only go back to D ALEMBERT S idea that 
the demonstration of the postulate can be connected with 
the definition of the straight line [cf. S 23]. 

Fourier, who regarded the distance between two points 
as a prime notion, proposed to define first the sphere; then 
the plane, as the locus of points equidistant from two 
given points; 3 then the straight line, as the locus of the 
points equidistant from three given points. This method 

* Cf. LAPLACE. Oeuvres, T. VI. Livre, V. Ch. V. p. 472. 

2 Cf. Seances de FEcolc normalt: De"bats, T. I. p. 2833 
(1795). This discussion was reprinted in Mathe"sis. T. IX. p. 139 
-141 (1883). 

3 This definition of the plane was given by LEIBNITZ about 
a century before. Cf. Opuscules et fragments inedits, edited by 
L. COUTURAT, p. 5545- (Paris, Alcan, 1903). 

Fourier and Legendre. cc 

of presenting the problem of the foundations of geometry 
agrees with the opinions adopted at a later date by other 
geometers, who made a special study of the question of 
this sense the discussion between FOURIER and MONGE finds 
a place among the earliest documents which refer to Non- 
Euclidean geometry? 

Adrien Marie Legendre [1752 1833!- 

27. The preceding geometers confined themselves to 
pointing out difficulties and to stating their opinions upon 
the Postulate. Legendre, on the other hand, attempted to 
transform it into a theorem. His investigations, scattered 
among the different editions of his Elements de Geometric 
[1794 1823], are brought together in his Reflexions sur 
differentes manures de dJmontrer la theorie des paralleles ou 
le ^heoreme sur la somme des trots angles du triangle. [Mem. 
Ac. Sc., Paris, T. XIII. 1833.] 

In jthe most interesting of his attempts, LEGENDRE, like 
SACCHERI, approaches the question from the side of the sum 
of the angles of a triangle, which sum he wishes to prove 
equal to two right angles. 

With this end in view, at the commencement of his work 
he succeeds in rejecting SACCHERI S Hypothesis of the Obtuse 
A?igle, since he establishes that the sum of the angles of any 
triangle is either less than {Hypothesis of the Acute Angle] or 
equal to [Hypothesis of the Right Angle} two right angles. 

We reproduce a neat and simple proof which he gives 
of this theorem: 

Let n equal segments A^A 2 , A 2 A^ . . . A H A n +i be taken 

1 To this we add that later memoirs and investigations 
showed that Fourier s definition also fails to build up the Eucli 
dean theory of parallels, without the help of the Fifth Postulate, 
or some other equivalent to it. 

c(5 II. The Forerunners of Non-Euclidean Geometry, 

one after the other on a straight line [Fig. 28]. On the same 
side of the line let n equal triangles be constructed, having 
for their third angular points B J3 2 . . .B n . The segments 
BiB 2 > B 2 B^... ,t-i B N , which join these vertices, are equal 
and can be taken as the bases of n equal triangles, B^A 2 B^ 

B, B 2 B 3 B* B^^A; -#- 

A H B H . The figure 
is completed by 
adding the triangle 

which is equal to 
the others. 

Let the angle BI of the triangle A^B^A 2 be denoted by 
P, and the angle A 2 of the consecutive triangle by a. 
Then P < a. 

In fact, if p >> a, by comparing the two triangles A^B^A 2 
and BiA 2 B 2 , which have two equal sides, we would deduce 

Further, since the broken line A^B^B 2 . . . B n +T. A-fi 
is greater than the segment A^An^ , 

But if n is taken sufficiently great, this inequality con 
tradicts the Postulate of Archimedes. 

Therefore A^A 2 is not greater than B^B 2 , 
and it follows that it is impossible that p ^> a. 

Thus we have p < a. 

From this it readily follows that the sum of the angles ot 
the triangle A^B^A 2 is less than or equal to two right angles. 

This theorem is usually, but mistakenly, called Legendre s 
First Theorem. We say mistakenly, because SACCHERI had 
already established this theorem almost a century earlier [cf 
p. 38] when he proved that the Hypothesis of the Obtuse 
Angle was false. 

Legendre s First Proof. tj 

The theorem usually called Legendre s Second Theorem 
was also given by SACCHERI, and in a more general form 
[cf. p. 29]. It is as follows: 

If the sum of the angles of a triangle is less than or 
equal to two right angles in only one triangle, it is respectively 
less than or equal to two right angles in every other triangle. 

We do not repeat the demonstration of this theorem, as 
it does not differ materially from that of SACCHERI. 

We shall rather show how LEGENDRE proves that the 
sum of the three angles of a tria?igle is equal to two right 

Suppose that in the triangle ABC [cf. Fig. 29] 
^A + 3^3 + ^C C< 2 right angles. 

A point D being taken on AB, the transversal DE is 
drawn, making the angle ADE 
equal to the angle B. In the quadri 
lateral DBCE the sum of the angles 
is less than 4 right angles. 

Therefore ^.AED^>^ACB. 
The angle E of the triangle ADE 
is then a perfectly definite [decreas 
ing] function of the side AD: or, A 
what amounts to the same thing, the 

length of the side AD is fully determined when we know the 
size (in right angles) of the angle E, and of the two fixed 
angles A^ B. 

But this result Legendre holds to be absurd, since the 
length of a line has not a meaning, unless one knows the unit 
of length to which it is referred, and the nature of the question 
does not indicate this unit in any way. 

In this way the hypothesis 

^A + ^B + <; C< 2 right angles 
is rejected, and consequently we have 

<; A + <$: + C= 2 right angles. 

e3 !! The Forerunners of Non-Euclidean Geometry. 

Also from this equality the proof of Euclid s Postulate 
follows easily. 

LEGENDRE S method is thus based upon LAMBERT S postu 
late, which denies the existence of an absolute unit segment. 

28. In another demonstration LEGENDRE makes use of 
the hypothesis: 

From any point whatever, taken within an angle, we can 
always draw a straight line which will cut the two arms of 
the angle? 

He proceeds as follows: 

Let ABC be a triangle, in which, if possible, the sum of 
the angles is less than two right angles. 

Let 2 right angles <^ A ^ B ^ C = a [the defect]. 
Find the point A , symmetrical to A, with respect to the 
side BC. ["cf. Fig. 30.] 

The defect of the new tri 
angle BCA is also a. In virtue 
of the hypothesis enunciated 
above, draw through A a 
transversal meeting the arms 
of the angle A in B^ and C r . 
It can easily be shown that the 
~ defect of the triangle AB^ C^ is 
the sum of the defects of the 
four triangles of which it is 
composed, [cf. also LAMBERT p. 46.] 

Thus this defect is greater than 2 a. 
Starting now with the triangle AB^d and repeating the 
same construction, we get a new triangle whose defect is 
greater than 4 a. 

i J. F. LORENZ had already used this hypothesis for the same 
purpose. Cf. Grundrifi der reincn und angewandten Mathemali * 
(Helmstedt, 1791). 

Legendre s Second Proof. 


After n operations of this kind a triangle will have been 
constructed whose defect is greater than 2 n a. 

But for n sufficiently great, this defect, 2 a, must be 
greater than 2 right angles {Postulate of Archimedes}, which 
is absurd. 

It follows that a = o, and <^ A + ^B + -^C = 2 
right angles. 

This demonstration is founded upon the Postulate of 
Archimedes. We shall now show how we could avoid using 
this postulate [cf. Fig. 31]. 

Let AB and HK be two straight lines, of which AB 
makes an acute angle, and HK a right angle, with AH. 

Fig. 31- 

Draw the straight line AB symmetrical to AB with re 
gard to AH. Through the point H there passes, in virtue of 
LEGENDRE S hypothesis, a line r which cuts the two arms of 
the angle BAB . If this line is different from HK, then also 
the line r, symmetrical to it with respect to AH, enjoys the 
same property of intersecting the arms of the angle. It fol 
lows that the line HK also meets them. 

Thus the line perpendicular to AH and a line making 
an acute angle with AH always meet. 

From this result the ordinary theory of parallels follows, 
and <^A + ^B + <; C= 2 right angles. 

In other demonstrations LEGENDRE adopts the methods 
of analysis and also makes an erroneous use of infinity. 

60 H- The Forerunners of Non-Euclidean Geometry. 

By these very varied investigations LEGENDRE believed 
that he had finally removed the serious difficulties surrounding 
the foundations of geometry. In substance, however, he 
added nothing new to the material and to the results ob 
tained by his predecessors. His greatest merit lies in the 
elegant and simple form which he was able to give to all his 
writings. For this reason they gained a wide circle of readers 
and helped greatly to increase the number of disciples of the 
new ideas, which at that time were beginning to be formed. 

Wolfgang Bolyai [1775-1856]. 

29. In this article we come to the work of the Hungarian 
geometer W. BOLYAI. His interest in the theory of parallels 
dates back to the time when he was a student at Gottingen 
[1796 99], and is probably due to the advice of KASTNER 
and of his friend, the young Professor of Astronomy, K. F. 
SEYFFER [1762 1822]. 

In 1804 he sent GAUSS, formerly one of his student 
friends at Gottingen, a Theoria Parallelarum, which contained 
an attempt at a proof of the existence of equidistant straight 
lines. 1 GAUSS showed that this proof was fallacious. BOLYAI 
however, did not on this account give up his study of Axiom 
XL, though he only succeeded in substituting for it others, 
more or less evident. In this way he came to doubt the possib 
ility of a demonstration and to conceive the impossibility 
of doing away with the Euclidean hypothesis. He asserted 
that the results derived from the denial of Axiom XI 
could not contradict the principles of geometry, since the 
law of the intersection of two straight lines, in its usual 

1 The Theoria Parallelarum was written in Latin. A German 
translation by ENGEL and SxACKEL appears in Math. Ann. Bd. 
XLIX. p. 168205 (1897). 

W. Bolyai s Postulate. 


form, represents a new datum, independent of those which 
precede it. 1 

WOLFGANG brought together his writings on the principles 
of mathematics in the work: Tentamen juventutem studiosam 
in elementa Matheseos [1832 33]; and in particular his in 
vestigations on Axiom XL, while in each attempt he pointed 
out the new hypothesis necessary to render the demon 
stration rigorous. 

A remarkable postidate to which WOLFGANG reduces 
EUCLID S is the following: 

Four points, not on a plane, always lie upon a sphere; 
or, what amounts to the same thing: A circle can always be 
drawn through three points not on a straight line. 2 

The Euclidean Postulate can be deduced from this as 
follows [cf. Fig. 32]: 

Let AA , BB be two straight lines, one of them being 
perpendicular to AB, and the other inclined to it at an acute 

If we take a point M on the seg 
ment AB between A and B, and 
the points M M" symmetrical to M 
with respect to the lines BB and 
AA, we obtain two points M , M" 
not in the same straight line with M. 
These three points M, M , M" lie 
on the circumference of a circle. Also 
the lines AA , BB must intersect, 
since they both pass through the cen 
tre of this circle. 

But from the fact that a line which is perpendicular to 





Fig. 32- 

1 Cf. STACKEL: Die Entdeckung der niehteuklidischen Geometrie 
dnrch J. Bolyai, Math. u. Naturw. Ber. aus Ungarn, Bd. XVII. (1901). 

2 Cf. W. BOLYAI : Kurzer Grundriss eines Versuchs etc., p. 46. 
(Maros Vdsarhely, 85(). 

62 II. The Forerunners of Non-Euclidean Geometry. 

another straight line and a line which cuts it at an acute angle 
intersect, it follows immediately that there can be only one 

Friedrich Ludwig Wachter [17921817]. 

30. When it had been seen that the Euclidean Postulate 
depends on the possibility of a circle being drawn through 
any three points not on a straight line, the idea at once sug 
gested itself that the existence of such a circle should be 
established as a preliminary to any investigation of parallels. 

An attempt in this direction was made by F. L. WACHTER. 

WACHTER, a student under GAUSS in Gottingen [1809], 
and Professor of Mathematics in the Gymnasium of Dantzig, 
had made several attempts at the demonstration of the Postu 
late. He believed that he had been successful, first in a letter 
to GAUSS [Dec., 1816], and later, in a tract, printed at Dantzig 
in 1817. * 

In this pamphlet he seeks to establish that given any four 
points in space, (not on a plane), a sphere will pass through 
them. He makes use of the following postulate : 

Any four points of space fully determine a surface [the 
surface of four points], and two of these surfaces intersect in a 
single line, completely determined by three points. 

There is no advantage in following the argument by 
means of which WACHTER seeks to prove that the surface of 
four points is a sphere, since he fails to give a precise defini 
tion of that surface in his tract. His deductions have thus 
only an intuitive character. 

On the other hand a passage in his letter of 1816 de 
serves special notice. It was written after a conversation with 
GAUSS, when they had spoken of an Anti- Euclidean Geometry. 
Tn this letter he speaks of the surface to which a sphere tends 

1 Demcnstratio axiomatis geometrici in Euclideis undecimi. 

Wachter and Thibaut. 63 

as its radius approaches infinity, a surface on the Euclidean 
hypothesis identical with a plane. He affirms that even in the 
case of the Fifth Postulate being false, there would be a geo 
metry on this surface identical with that of the ordinary plane. 
This statement is of the greatest importance as it con 
tains one of the most remarkable results which hold in the 
system of geometry, corresponding to SACCHERI S Hypo 
thesis of the Acute Angle [cf. LOBATSCHEWSKY, 40].* 

Bernhard Friedrich Thibaut [1775 1832]. 

30 (bis). One other erroneous proof of the theorem that the 
sum of the angles of a triangle is equal to two right angles should 
be mentioned, since it has recently been revived in English textbooks, 
and to some extent received official sanction. It depends upon 
the idea of direction, and assumes that translation and rotation are 
independent operations. It is due to THIBAUT (Grundrifi der reinen 
Mathematik, 2. Aufl., Gottingen, 1809). GAUSS refers to this "proof" 
in his correspondence with SCHUMACHER, and shows that it involves 
a proposition which not only needs proof, but is, in essence, the 
very proposition to be proved. THIBAUT argued as follows : 2 

"Let ABC be any triangle whose sides are traversed in order 
from A along AB, BC, CA. While going from A to B we always 
gaze in the direction ABb (AB being produced to b], but do not 
turn round. On arriving at B we turn from the direction Bb by a 
rotation through the angle bBC, until we gaze in the direction BCc. 
Then we proceed in the direction BCc as far as C, where again 
we turn from Cc to CAa through the angle cCA\ and at last arriving 
at A, we turn from the direction Aa to the first direction AB 
through the external angle aAB. This done, we have made a 
complete revolution, just as if, standing at some point, we had 
turned completely round; and the measure of this rotation is 2 IT. 
Hence the external angles of the triangle add up to 2 IT, and the 
internal angles A + B -f- C = IT. Q. E. D." 

i With regard to WACHTER, cf. P. STACKEL: Friedrich Ltidwig 
Wachter, ein Beitrag zur Geschichte der nichteuklidischen Geometric. 
Math. Ann. Bd. LIV. p. 4985. (1901). In this article are reprinted 
WACHTER S letters upon the subject and the tract of 1817 referred 
to above. 

a [For further discussion of this "proof" see W. B. FRANK- 
LAND S Theories of Parallelism, (Camb. Univ. Press, 1910), from which 
this version is taken, and HEATH S Euclid, Vol. I., p. 321.] 

Chapter III. 

The Founders of Non-Euclidean Geometry. 
Carl Friederich Gauss [1777 1855]. 

31. Twenty centuries of useless effort, and in particular 
the last unsuccessful investigations on the Fifth Postulate, con 
vinced many of the geometers, who flourished about the be 
ginning of last century, that the final settlement of the theory 
of parallels involved a problem whose solution was impossible. 
The Gottingen school had officially declared the necessity 
of admitting the Euclidean hypothesis. This view, expressed 
by KLUGEL in his Conatuum [cf. p. 44] was accepted and sup 
ported by his teacher, A. G. KASTNER, then Professor in the 
University of Gottingen. 1 

Nevertheless keen interest was always taken in the 
subject; an interest which still continued to provide those 
who sought for a proof of the postulate with fruitless labour, 
and led finally to the discovery of new systems of geometry. 
These, founded like ordinary geometry on intuition, extend 
into a far wider field, freed from the principle embodied in 
the Euclidean Postulate. 

How difficult was this advance towards the new order 
of ideas will be clear to any one who carries himself back to 
that period, and remembers the trend of the Kantian Philo 
sophy, then predominant. 

32. GAUSS was the first to have a clear view of a 
geometry independent of the Fifth Postulate, but this re- 

Cf. ENGEL u. STACKEL: Th. der P. p. 139142. 

Gauss and W. Bolyai. 65 

mained for quite fifty 1 years concealed in the mind of the 
great geometer, and was only revealed after the works of 
LOBATSCHEWSKY [1829 30] and J. BOLYAI [1832] appeared. 

The documents which allow an approximate reconstruct 
ion of the lines of research followed by GAUSS in his work 
on parallels, are his correspondence with W. BOLYAI, OLBERS, 
two short articles in the Gott. gelehrten Anzeigen [1816, 1822]; 
and some notes found among his papers, [i83i]. 2 

Comparing the various passages in GAUSS S letters, we 
can fix the year 1792 as the date at which he began his Med 
itations . 

The following portion of a letter to W. BOLYAI [Dec. 1 7, 
1799] proves that GAUSS, like SACCHERI and LAMBERT before 
him, had attempted to prove the truth of Postulate V. by as 
suming it to be false. 

( As for me, I have already made some progress in my 
work. However the path I have chosen does not lead at 
all to the goal w^iich we seek, and which you assure me you 
have reached. 3 / It seems rather to compel me to doubt the 
truth of geometry itself, j 

It is true that I have come upon much which by most 
people would be held to constitute a proof: but in my eyes 
it proves as good as nothing. For example, if one could 
show that a rectilinear triangle is possible, whose area would 
be greater than any given area, then I would be ready to 
prove the whole of geometry absolutely rigorously. 

Most people would certainly let this stand as an Axiom; 
but I, no! It would, indeed, be possible that the area might 

* [It would be more correct to say over thirty.] 

2 Cf. GAUSS, Werke, Bd. VIII. p. 157268. 

3 It is to be remembered that W. BOLYAI was working at 
this subject in Gottingen and thought he had overcome his diffi 
culties. Cf. S 2 9- 


66 HI. The Founders of Non-Euclidean Geometry. 

always remain below a certain limit, however far apart the 
three angular points of the triangle were taken. 

In 1804, replying to W. BOLYAI on his Theoria parall- 
elarum, he expresses the hope that the obstacles by which 
their investigations had been brought to a standstill would 
finally leave a way of advance open. 1 

From all this, STACKEL and ENGEL, who collected and 
verified GAUSS S correspondence on this subject, come to the 
conclusion that the great geometer did not recognize the 
existence of a logically sound Non-Euclidean geometry by 
intuition or by a flash of genius : that, on the contrary, he 
had spent upon this subject many laborious hours before he 
had overcome the inherited prejudice against it. 

Did GAUSS, when he began his investigations, know the 
writings of SACCHERI and LAMBERT? What influence did they 
exert upon his work? Segre, in his Congetture, already re 
ferred to [p. 44 note 2], remarks that both GAUSS and W. 
BOLYAI, while students at Gottingen, the former from 1795 
98, the later from 1796 99, were interested in the theory 
of parallels. It is therefore possible that, through KASTNER 
and SEYFFER, who were both deeply versed in this subject 
they had obtained knowledge both of the Euclides ab omni 
naevo vindicatus and of the Theorie der Parallellinien. But 
the dates of which we are certain, although they do not con 
tradict this view, fail to confirm it absolutely. 

33- To this first period of GAUSS S work, after 1813 
there follows a second. Of it we obtain some knowledge 
chiefly from a few letters, one written by WACHTER to GAUSS 
[1816]; others sent by GAUSS to GERLING [1819], TAURINUS 
[1824] and SCHUMACHER [1831]; and also from some notes 
found among GAUSS S papers. 

* [It should be noticed that these efforts were still directed 
towards proving the truth of Euclid s postulate.] 

Gauss s "Meditations". 67 

These documents show us that GAUSS, in this second 
period, had overcome his doubts, and proceeded with his de 
velopment of the fundamental theorems of a new geometry, 
which he first calls Anti- Euclidean [cf.WACHTER s letter quoted 
on p. 62]; then Astral Geometry [following SCHWEIKART, cf. 
p. 76]; and finally, Non-Euclidean [cf. letter to SCHUMACHER]. 
Thus he became convinced that the Non-Euclidean Geometry 
did not in itself involve any contradiction, though at first 
sight some of its results had the appearance of paradoxes 
[letter to SCHUMACHER, July 12, 1831]. 

However GAUSS did not let any rumour of his opinions 
get abroad, being certain that he would be misunderstood. 
[He was afraid of the clamour of the Boeotians; letter to BESSEL, 
Jan. 27, 1829]. Only to a few trusted friends did he reveal 
something of his work. When circumstances compel him to 
write to TAURINUS [1824] on the subject, he begs him to 
keep silence as to the information which he imparted to him. 

The notes found among GAUSS S papers contain two 
brief synopses of the new theory of parallels, and probably 
belong to the projected exposition of the Non-Euclidean Geo 
metry, with regard to which he wrote to SCHUMACHER [on 
May 17, 1831]: In the last few weeks I have begun to put 
down a few of my own Meditations, which are already to 
some extent nearly 40 years old. These I had never put in 
writing, so that I have been compelled three or four times 
to go over the whole matter afresh in my head. Also I wished 
that it should not perish with me. 

34. GAUSS defines parallels as follows: r 
Jf the coplanar straight lines AM, BN, do not intersect 
each other, while; on the other hand, every straight line through 

1 [In this section upon GAUSS S work on Parallels fuller use 
has been made of the material in his Collected Works (GAUSS, 
Werke, Bd. VIII, p. 2029)]. 


68 HI. The Founders of Non-Euclidean Geometry. 

A between AM and AB cuts BN } then AM is said to beparal- 

He supposes a straight 
line passing through A, to 
start from the position AB, 
and then to rotate continu 
ously on the side towards 
^"^ which BN is drawn, till it 
reaches the position AC, in 
BA produced. This line be 
gins by cutting ^TVand in the 
Fig 33 end it does not cut it. Thus 

there can be one and only 

one position, separating the lines which intersect BN horn 
those which do not intersect it. This must be \hzfirst of the 
lines, which do not cut BN: and thus from our definition it 
is the parallel AM\ since there can obviously be no last line 
of the set of lines which intersect BN. 

It will be seen in what way this definition differs from 
EUCLID S. If EUCLID S Postulate is rejected, there could be dif 
ferent lines through A, on the side towards which BN is 
drawn, which would not cut BN. These lines would all be 
parallels to BN according to EUCLID S Definition. In GAUSS S 
definition only the first of these is said to be parallel 

Proceeding with his argument GAUSS now points out 
that in his definition the starting points of the lines AM and 
BN are assumed, though the lines are supposed to be pro 
duced indefinitely in the directions of AM and BN. 

L He proceeds to show that the parallelism of the line 
AM to the line BN is independent of the points A and B, pro 
vided the sense in which the lines are to be produced indefinitely 
remain the same. 

It is obvious that we would obtain the same parallel AM 

Gauss s Theory of Parallels. 

if we kept A fixed and took instead of B another point B 
on the line BN, or on that line produced backwards. 

It remains to prove that if AM is parallel to BNioi the 
point A, it is also the parallel to BNtei any point upon AM, 
or upon AM produced backwards. 

Instead of A [Fig. 34] take another starting point A upon 
AM. Through A , between 
A B and A M, draw the line 
A P in any direction. B 

Through Q, any point on 
A P, between A and P, draw 
the line AQ. 

Then, from the definition, A 
AQ must cut BN, so that it 
is clear QP must also cut 

Thus A A Mis the first of 

the lines which do not cut BN, and AM is parallel to BN. 

Again take the point A upon AM produced backwards 

[Fig- 35]- 


Fig. 34. 

Fig. 35- 

Draw through A, between AB and AM, the line A P 
in any direction. 

Produce AP backwards and upon it take any point Q. 
Then, by the definition, QA must cut BN, for example, 


III. The Founders of Non-Euclidean Geometry. 

in R. Therefore A P lies within the closed figure AARB, 
and must cut one of the four sides A A, AR^ RB^ and BA. 
Obviously this must be the third side RB, and therefore 
AM is parallel to BN. 

II. The Reciprocity of the Parallelism can also be estab 

In other words, if AM is parallel to BN^ then BN is 
also parallel to AM. 

GAUSS proves this result as follows: 

From any point B upon BN draw BA perpendicular to 
AM. Through B draw any line BN between BA and BN. 

At B, on the same side of AB as BN, make 

There are two possible cases: 

Case (i), when BC cuts AM [cf. Fig. 36]. 
Case (ii), when BC does not cut AM [cf. Fig. 37]. 

Case (i). Let BC cut AM in D. Take AE = AD> and 
join BE. Make ^BDF=^BED. 

Since AM is parallel to BN, DF must cut &M, for 
example, in G. &W 

From EM cut off EH equal to DG. 

Then, in the triangles BEH and BDG, it follows that 

Gauss s Theory of Parallels (contd.). 

Therefore < EBD = <^ HBG. 

But < EBD = <; N BN. 
Therefore BN* and BH coincide, and BN must cut 


But BN 1 is any line through B y between BA and 
Therefore BN is parallel to AM. 


fig- 37- 

Case (ii). In this case let D be any arbitrary point upon 
AM. Then with the same argument as above, 
< EBD = ^ GBH. 

But ^c ^?z> < <: ^c. . 

Therefore ^^Z> < 

Therefore ^C GBH < 

Therefore BN must cut AM. 

But AV is any line through B, between BA and BN. 

Therefore BN is parallel to AM. 

Thus in both cases we have proved that if AM is parallel 
to BN, then ^^Vis parallel to AM. 1 

The next theorem proved by GAUSS in this synopsis is 
as follows : 

[ J GAUSS S second proof of this theorem is given in the German 
translation. However it will be found that in it he assumes that BC 
cuts AM, and to prove this the argument used above is necessary.] 

III. The Founders of Non-Euclidean Geometry. 

Fig. 38. 

III. If the line (i) is parallel to the line (2) and to the 
line (3), then (2) and (3) are parallel to each other. 

Case (i). Let the line (i) lie between (2) and (3) [cf. 

Fig- 38]. 

Let A and B be two points on (2) and (3), and let AB 
cut (i) in C. 

Through A let an arbitrary line AD be drawn between 

AB and (2). Then it must 
cut (i), and on being pro 
duced must also cut (3). 

Since this holds for every 
line such as AD, (2) is 
parallel to (3). 

Case (ii). Let the line 
(i) be outside both (2) and 
(3), and let (2) lie between 
(i) and (3) [cf. Fig. 39]. 
If (2) is not parallel to (3), through any point chosen at 
random upon (3); a line different from (3) can be drawn 
which is parallel to (2). 

This, by Case (i), is also par 
allel to (i), which is absurd. 
This short Note on Parall 
els closes with the theorem 
that // two lines AM and BN 

are parallel, these lines produced 
backwards cannot meet. 
From all this it is evident that the parallelism of GAUSS 
means parallelism in a given sense. Indeed his definition of 
parallels deals with a line drawn from A on a definite side of 
the transversal AB\ e. g., the ray drawn to the right, so that 
we might speak of AM as the parallel to BN towards the right. 
The parallel from A to BN towards the left is not necessari 
ly AM. If it were, we would obtain the Euclidean hypothesis. 

Fig. 39- 

Corresponding Points. 


The two lines, in the third theorem, which are each pa 
rallel to a third line, are thus both parallels in the same sense 
(both left-hand, or both right-hand parallels). 

In a second memorandum on parallels, GAUSS goes over 
the same ground, but adds the idea of Corresponding Points 
on two parallels AA , BB . Two points A, B arc said to corre 
spond, when AB makes equal internal angles with the parallels 
on the same side [cf. Fig. 40]. 


Fig. 40. 

Fig. 41. 

With regard to these Corresponding Points he states the 
following theorems: 

(i) If A, B are two corresponding points upon two paral 
lels, and M is the middle point of AB, the line MN, perpen 
dicular to AB, is parallel to the two given lines, and every 
point on the same side of MN as A is nearer A than B. 

(ii) If A) B are two corresponding points upon the 
parallels(i] and (2), and A , B two other corresponding points 
on the same lines, then AA = BB\ and conversely. 

(iii) If A y B, C are three points on the parallels (i), (2) 
and (3), such that A and B, B and C } correspond, then A and 
C also correspond. 

74 HI. The Founders of Non-Euclidean Geometry. 

The idea of Corresponding Points, when taken in con 
nection with three lines of a pencil (that is, three concurrent 
lines [cf. Fig. 41] allows us to define the circle as the locus of 
the points on the lines of a pencil which correspond to a given 
point. But this locus can also be constructed when the lines 
of the pencil are parallel. In the Euclidean case the locus 
is a straight line : but putting aside the Euclidean hypothesis, 
the locus in question is a line, having many properties in 
common with the circle, but yet not itself a circle. Indeed if 
any three points are taken upon if, a circle cannot be drawn 
through them. This line can be regarded as the limiting case 
of a circle, when its radius becomes infinite. In the Non- 
Euclidean geometry of LOBATSCHEWSKY and BOLYAI, this locus 
plays a most important part, and we shall meet it there under 
the name of the Horocycle. 1 

This work GAUSS did not need to complete, for in 1832 
he received from WOLFGANG BOLYAI a copy of the work of 
his son JOHANN on Absolute Geometry. 

From letters before and after the date at which he 
interrupted his work, we know that GAUSS had discovered in 
his geometry an Absolute Unit of Length [cf. LAMBERT and 
LEGENDRE], and that a constant k appeared in his formulae, 
by means of which all the problems of the Non-Euclidean 
Geometry could be solved [letter to Taurinus, Nov. 8, 

Speaking more fully of these matters in 1831 [letter to 

1 [LOBATSCHEWSKY : Grenzkreis, Courbe-limile or Horicyde. BOL- 
YAI; Parazykl, L-linie. 

It is interesting to notice that GAUSS, even at this date, 
seems to have anticipated the importance of the Horocycle. The 
definition of Corresponding Points and the statement of their 
properties is evidently meant to form an introduction to the dis 
cussion of the properties of this curve, to which he seems to have 
given the name Trope. ] 

The Perimeter of a Circle. 75 

SCHUMACHER], he gave the length of the circumference of a 
circle of radius r in the form 

With regard to k, he says that, if we wish to make the new 
geometry agree with the facts of experience, we must suppose 
k infinitely great in comparison with all known measurements. 
For k oo , GAUSS S expression takes the usual form 
for the perimeter of a circle. x The same remark holds for the 
whole of GAUSS S system of geometry. It contains EUCLID S 
system, as the limiting case, when k = 00 . 2 

Ferdinand Karl Schweikart [1780 1859]. 

35. The investigations of the Professor of Jurispru 
dence, F. K. SCHWEIKART^ date from the same period as 
those of GAUSS, but are independent of them. In 1807 he 
published Die Theorie der Parallellinien nebst dem Vorschlage 
ihrer Verbannung aus der Geometrie. Contrary to what one 
might expect from its title, this work does not contain a 
treatment of parallels independent of the Fifth Postulate, 
but one based on the idea of the parallelogram. 

But at a later date, SCHWEIKART, having discovered a 
new order of ideas, developed a geometry independent of 
Euclid s hypothesis. When in Marburg in December, 1818, 
he handed the following memorandum to his colleague GER- 
LING, asking him to communicate it to GAUSS and obtain his 
opinion upon it: 

1 To show this we need only use the exponential series. 

2 For other investigations by GAUSS, cf. Note on p. 90. 

3 He studied law at Marburg and from 179698 attended the 
lectures on Mathematics given in that University by Professor J. K. 
F. HAUFF, the author of various memoirs on parallels, cf. Th. der 
P. p- 243- 

in. The Founders of Non-Euclidean Geometry. 


There are two kinds of geometry a geometry in the 
strict sense the Euclidean; and an astral geometry [astra- 
lische Grofienlehre]. 

Triangles in the latter have the property that the sum 
of their three angles is not equal to two right angles. 

This being assumed, we can prove rigorously: 

a) That the sum of the three angles of a triangle is less 
than two right angles; 

b) that the sum becomes ever less, the greater the area 
of the triangle; 

c) that the altitude of an isosceles right-angled triangle 
continually grows, as the sides increase, but it can 
never become greater than a certain length, which 
I call the Constant. 

Squares have, therefore, the following form [Fig. 42]. 
If this Constant were for us the Radius of the Earth, 
(so that every line drawn in the 
universe from one fixed star 
to another, distant 90 from the 
first, would be a tangent to the 
surface of the earth), it would be 
infinitely great in comparison with 
the spaces which occur in daily 

The Euclidean geometry holds 
only on the assumption that the 
Constant is infinite. Only in this 
case is it true that the three angles of every triangle are equal 
to two right angles: and this can easily be proved, as soon 
as we admit that the Constant is infinite. x 

SCHWEIKART S Astral Geometry and GAUSS S Non-Euclid- 

Fig. 42. 

Schweikart s Work. 


can Geometry exactly correspond to the systems of SAC- 
CHERI and LAMBERT for the Hypothesis of the Acute Angle. 
Indeed the contents of the above memorandum can be ob 
tained directly from the theorems of SACCHERI, stated in 
KLOGEL S Conatuum, and from LAMBERT S Theorem on the 
area of a triangle. Also since SCHWEIKART in his Theorie of 
1807 mentions the works of the two latter authors, the direct 
influence of LAMBERT, and, at least, the indirect "influence of 
SACCHERI upon his investigations are established. 2 

In March, 1819 GAUSS replied to GERLING with regard 
to the Astral Geometry. He compliments SCHWEIKART, and 
declares his agreement with all that the sheet of paper sent 
to him contained. He adds that he had extended the Astral 
Geometry so far that he could completely solve all its pro 
blems, if only SCHWEIKART S Constant were given. In con 
clusion, he gives the upper limit for the area of a triangle 
in the form 3 

[log hyp (i -f- V" JJ 2 
SCHWEIKART did not publish his investigations. 

Franz Adolf Taurinus [1794 1874]. 
36. In addition to carrying on his own investigations 
on parallels, SCHWEIKART had persuaded [1820] his nephew 
TAURINUS to devote himself to the subject, calling his atten- 

1 Cf. GAUSS, Werke, Bd. VIII, p. 180 181. 

2 Cf. SEGRE S Congetture^ cited above on p. 44. 

3 The constant which appears in this formula is SCHWEIKART S 
Constant C, not GAUSS S constant <, in terms of which he expressed 
the length of the circumference of a circle, (cf. p. 75). The two 
constants are connected by the following equation: 

"log (I-T-V2) 

78 EL Tte Founders of Non-Euclidean Geometry. 

tion to the Astral Geometry, and to GAUSS S favourable ver 
dict upon it. 

TAURINUS appears to have taken up the subject seriously 
for the first time in 1824, but with views very different from 
his uncle s. He was then convinced of the absolute truth of 
the Fifth Postulate, and always remained so, and he cherish 
ed the hope of being able to prove it. Failing in his first at 
tempts, under the influence of GAUSS and SCHWEIKART, he 
again began the study of the question. In 1 8 2 5 he publish 
ed a Theorie der Parallellinien, containing a treatment of the 
subject on Non-Euclidean lines, the rejection of the Hypothesis 
of the Obtuse Angle, and some investigations resembling those 
of SACCHERI and LAMBERT on the Hypothesis of the Acute 
Angle. He found in this way SCHWEIKART S Constant, which 
he called a Parameter. He thought an absolute unit of 
length impossible, and concluded that all the systems, corre 
sponding to the infinite number of values of the parameter, 
ought to hold simultaneously. But this, in its turn, led to con 
siderations incompatible with his conception of space, and 
thus TAURINUS was led to reject the Hypothesis of the Acute 
Angle while recognising the logical compatibility of the propo 
sitions which followed from it. 

In the next year TAURINUS published his Geometriae Pri- 
ma Elementa [Cologne, 1826], in which he gave an improved 
version of his researches of 1825. This work concludes with 
a most important appendix, in which the author shows how 
a system of analytical geometry could be actually constructed 
on the Hypothesis of the Acute Angle.* 

With this aim TAURinus starts from the fundamental for 
mula of Spherical Trigonometry 

i For the final influence of SACCHERI and LAMBERT upon TAU 
RINUS, cf. SEGRE s Congetture, quoted above on p. 44. 

The Work of Taurinus. 

cos -^ = cos ^ cos - + sin sin k cos 


In it he transforms the real radius k into the imaginary radius 

ik. Using the notation of the hyperbolic functions, we thus 


(i) cosh -,- = cosh -j- cosh -r- sinh sinh ~ cos A. 

K K k k k 

This is the fundamental formula of the Logarithmic- 
Spherical Geometry \logarithmisch-sphdrischen Geometrie\ of 

It is easy to show that in this geometry the sum of the 
angles of a triangle is less than 180. For simplicity we take 
the case of an equilateral triangle, putting a=b=c in (i). 

Solving, for cos A, we obtain 

(i*) cos A = - 


But sech -, < i. 


Therefore cos A ^> x / 2 . 

Thus A is less than 60, and the sum of the angles of 
the triangle is less than 1 80. 

It is instructive to note, that, from (i*). 
Lt. (cos A) = V 2 . 

a - o 

So that in the limit when a becomes zero, A is equal to 60. 
Therefore, in the log -spherical geometry, the sum of the angles 
of a triangle tends to 180 when the sides tend to zero. 
We may also note that from (i*) 

so that in the limit when k is infinite, A is equal to 60. There 
fore, when the constant k tends to infinity, the angles of the 
equilateral triangle are each equal to 60, as in the ordinary 

8o HI. The Founders of Non-Euclidean Geometry. 

More generally, using the exponential forms for the hy 
perbolic functions, it will be seen that in the limit when k is 
infinite (i) becomes 

a 2 = b 2 + c z 2bc cos A, 
the fundamental formula of Euclidean Plane Trigonometry. 

37. The second fundamental formula of Spherical 

cos A = cos B cos C + sin B sin C cos ~ k , 

by simply interchanging the cosine with the hyperbolic cosine, 
gives rise to the second fundamental formula of the log -spher 
ical geometry: 

(2) cos A = cos B cos C 4- sin B sin C cosh -,. 
For A = o and C= 90, we have 

The triangle corresponding to this formula has one angle 
zero and the two sides containing it are infinite and parallel 
[asymptotic]. [Fig. 43.] The angle B, between the side which 


Fig. 43- 

is parallel and the side which is perpendicular to CA, is seen 
from (3) to be a function of a. From this onward we can 
call it the Angle of Parallelism for the distance a [cf. LOBAT- 
SCHEWSKY, p. 87]. 

For B = 45, the segment -BC, which is given by (3), is 
SCHWEIKART S Constant [cf. p. 76]. Thus, denoting it by P, 

The Angle of Parallelism. 

cosh - k = V 2, 

from which, solving for k, we have 


log (I +V2~) 

This relation connecting the two constants P and k was 
given by TAURINUS. The constant k is the same as that em 
ployed by GAUSS [cf. p. 75] in finding the length of the cir 
cumference of a circle. 

38. TAURINUS deduced other important theorems in 
the log.-spherical geometry by further transformations of the 
formulae of Spherical Trigonometry, replacing the real radius 
by an imaginary one. 

For example, that the area of a triangle is proportional 
to its defect [LAMBERT, p. 46] : 

that the superior limit of that area is 

nP* r ~ 

^r [GAUSS, p. 77]; 
[log(i-fV2)] 2 L 

that the length of the circumference of a circle of radius r is 

2TT/& sinh -j [GAUSS, p. 75]; 
that the area of a circle of radius r is 

2ir/ 2 (cosh -J i); 


that the area of the surface of a sphere and its volume, are 

4ir a sinh 2 -,, 

and 2TT/3 (sinh k cosh k -^). 

We shall not devote more space to the different analyt- 


82 HI- The Founders of Non-Euclidean Geometry. 

ical developments, since a fuller discussion would cast no 
fresh light upon the method. However we note that the 
results of TAURINUS confirm the prophecy of LAMBERT on 
the Third Hypothesis [cf. p. 50], since the formulae of the 
log.-spherical geometry, interpreted analytically, give the fun 
damental relations between the elements of a triangle traced 
upon a sphere of imaginary radius. 1 

To this we add that TAURINUS in common with LAMBERT 
recognized that Spherical Geometry corresponds exactly to 
the system valid in the case of the Hypothesis of the Obtuse 
Angle: further that the ordinary geometry forms a link be 
tween spherical geometry and the log, -spherical geometry. 

Indeed, if the radius k passes continuously from the real 
domain to the purely imaginary one, through infinity, we pro 
ceed from the spherical system to the log. -spherical system, 
through the Euclidean. 

Although TAURINUS, as we have already remarked, ex 
cluded the possibility that a log.-spherical geometry could be 
valid on the plane, the theoretical interest, which it offers, 
did not escape his notice. Calling the attention of geo 
meters to his formulae, he seemed to prophecy the existence 

1 At this stage it should be remarked that LAMBERT, simul 
taneously with his researches on parallels, was working at the tri 
gonometrical functions with an imaginary argument, whose connection 
with Non-Euclidean Geometry was brought to light by TAURINUS. 
Perhaps LAMBERT recognised that the formulae of Spherical Trig 
onometry were still real, even when the real radius was changed 
in a purely imaginary one. In this case his prophecy with regard 
to the Hypothesis of the Acute Angle (cf. p. 50) would have a firm 
foundation in his own work. However we have no authority for 
the view that he had ever actually compared his investigations on 
the trigonometrical functions with those on the theory of parallels. 
Cf. P. STACKEL: Bemcrkungcn su Lamberts Theorie der Parallellinien. 
Uiblioteca Math. p. 107110. (1899). 

Some Conclusions by Taurinus. 33 

of some concrete case in which they would find an inter 
pretation. 1 

i The important service rendered by SCHWEIKART and TAU 
RINUS towards the discovery of the Non-Euclidean Geometry was 
recognised and made known by ENGEL and STACKEL. In their 
Th. der P., they devote a whole chapter to those authors, and 
quote the most important passages in TAURINUS writings, besides 
some letters which passed between him, GAUSS and SCHWEIKART. 
Cf. STACKEL: Franz Adolf Taurinus, Abhandl. zur Geschichte der 
Math., IX, p. 397427 (1899). 


Chapter IV. 

The Founders of Non-Euclidean Geometry 
. (Contd.). 

Nicolai Ivanovitsch Lobatschewsky [1793 1856]. 1 

39. LOBATSCHEWSKY studied mathematics at the Uni 
versity of Kasan under a German J. M. C. BARTELS [1769 
1836], who was a friend and fellow countryman of GAUSS. 
He took his degree in 1813 and remained in the University, 
first as Assistant, and then as Professor. In the latter position 
he lectured upon mathematics in all its branches and also 
upon physics and astronomy. 

As early as 1815 LOBATSCHEWSKY was working at paral 
lels, and in a copy of his notes for his lectures [1815 17] 
several attempts at the proof of the Fifth Postulate, and 
some investigations resembling those of LEGENDRE have been 

However it was only after 1823 that he had thought of 
the Imaginary Geometry. This may be inferred from the 
manuscript for his book on Elementary Geometry, where he 
says that we do not possess any proof of the Fifth Postulate, 
but that such a proof may be possible- 2 

1 For historical and critical notes upon LOBATSCHEWSKY we 
refer once and for all to F. ENGEL S book: N. I. I.OBATSCHEFSKIJ : 
Zwei geometrische Abhandlnngen aus dem Russischen itbersetzt mit 
Anmerkungen und mit einer Biographie des Vcrjasscrs. (Leipzig, 
Teubner, 1899). 

2 [This manuscript had been sent to St. Petersburg in 1823 
to he published. However it was not printed, and it was dis- 

Lobatschewsky s Works. 85 

Between 1823 and 1825 LOBATSCHEWSKY had turned 
his attention to a geometry independent of Euclid s hypothe 
sis. The first fruit of his new studies is the Exposition suc- 
cincte des principes de la geometric avec une demonstration ri- 
goureusedu thtoreme des parables, read on 12 [24] Feb., 1826, 
to the Physical Mathematical Section of the University of 
Kasan. In this "Lecture", the manuscript of which has 
not been discovered, LOBATSCHEWSKY explains the prin 
ciples of a geometry, more general than the ordinary geo 
metry, where two parallels to a given line can be drawn 
through a point, and where the sum of the angles of a tri 
angle is less than two right angles \TheHypothesis of the Acute 
Angle of SACCHERI and LAMBERT]. 

Later, in 1829 30, he published a memoir On the Prin 
ciples of Geometry* containing the essential parts of the 
preceding "Lecture", and further applications of the new 
theory in analysis. In succession appeared the Imaginary 
Geometry [i835], 2 New Principles of Geometry, with a Com- 

covered in the archives of the University of Kasan in 1898. It 
is clear from some other remarks in this work that he had made 
further advance in the subject since 1815 17. He was now con 
vinced that all the first attempts at a proof of the Parallel Postulate 
were unsuccessful, and that the assumption that the angles of a 
triangle could depend only on the ratio of the sides and not upon 
their absolute lengths was unjustifiable (cf. ENGEL, loc.cit. p. 36970).] 

1 Kasan Bulletin, (1829 1830). Geometrical Works of Lobat- 
scheivsky (Kasan 1883 1886), Vol.1 p. I 67. German translation 
by F. ENGEL p. I 66 of the work referred to on the previous page. 

Where the titles are given in English we refer to works pub 
lished in Russian. The Geometrical Works of Lobatschewsky contain 
two parts; the first, the memoirs originally published in Russian; 
the second, those published in French or German. It will be seen 
below that of the works in Vol. i. several translations are now 
to be had. 

2 The Scientific Publications of the University of Kasan (1835). 
Geometrical Works, Vol. I, p. 71120. German translation by 

86 IV. The Founders of Non-Euclidean Geometry (Contd.). 

plete Theory of Parallels [1835 38] x , the Applications of the 
Imaginary Geometry to Some Integrals [1836]*, then the 
Geomtirie Imaginaire [183 7] 3, and in 1840, a small book 
containing a summary of his work, Geometrische Unter- 
suchungen zur Theorie der Parallellinien, 4 written in German 
and intended by LOBATSCHEWSKY to call the attention of 
mathenijaticans to his researches. Finally, in 1855, a year 
before his death, when he was already blind, he dictated and 
published in Russian and French a complete exposition of his 
system of geometry under the title: Pangeometrie ou precis 
de geomforie fondee sur une theorie gdncrale et rigoureuse des 
paralleles. 5 

40. Non-Euclidean Geometry, just as it was conceived 
by GAUSS and SCHWEIKART in 1816, and studied as an ab- 

H. LIEBMANN, with Notes. Abhandlungen zur Geschichte der Mathe- 
matik, Bd. XIX, p. 350 (Leipzig, Teubner, 1904). 

1 Scientific Publications of the University of Kasan (183538). 
Geom. Works. Vol. I: p. 219 486. German translation by F. ENGEL, 
p. 67 235 of his work referred to on p. 84. English translation 
of the Introduction by G. B. HALSTED, (Austin, Texas, 1897). 

2 Scientific Publications of the University of Kasan. (1836). 
Geom. Works, Vol. I, p. 1 21 2 1 8. German translation by H. LlEB- 
MANN; loc. cit: p. 51 130. 

3 CRELLE S Journal, Bd. XVII, p. 295320. (1837). Geom. 
Works, Vol. II, p. 581613. 

4 Berlin (1840). Geom. Works, Vol. II, p. 553578. French 
translation by J. HOUEL in Mem. de Bourdeaux, T. IV. (1866), and 
also in Recherches gcometriques sur la theorie des paralleles (Paris, Her 
mann, 1900). English translation by G. B. HALSTED, (Austin, 
Texas, 1891). Facsimile reprint (Berlin, Mayer and Miiller, 1887). 

5 Collection of Memoirs by Professors of the Royal University of 
Kasan on the tjO th anniversary of its foundation. Vol. I, p. 279 340. 
(1856). Also in Geom. Works, Vol. II, p. 617680. In Russian, in 
Scientific Publications of the University of Kasan, (1855). Italian 
translation, by G. BATTAGLINI, in Giornale di Mat. T. V. p. 273 336, 
(1867). German translation, by H. LIEBMANN, Ostwald s Klassikcr 
der exakten Wissenschaften, Nr, 130 (Leipzig, 1902). 

Lobatschewsky s Theory of Parallels. 37 

stract system by TAURINUS in 1826, became in 1829 30 
a recognized part of the general scientific inheritance. 

To describe, as shortly as possible, the method followed 
by LOBATSCHEWSKY in the construction of the Imaginary Geo 
metry or Pangeometry, let us glance at his Geometrische Unter- 
suchungen zur Theorie der Parallellinien of 1840. 

In this work LOBATSCHEWSKY states, first of all, a group 
of theorems independent of the theory of parallels. Then he 
considers a pencil with vertex 
A, and a straight line BC, in 
the plane of the pencil, but 
not belonging to it. Let AD 
be the line of the pencil which 

is perpendicular to BC, and ~ hr 

AE that perpendicular to 
AD. In the Euclidean system 

this latter line is the only line which does not intersect BC. 
In the geometry of LOBATSCHEWSKY there are other lines of the 
pencil through A which do not intersect BC. The non-inter 
secting lines are separated from the intersecting lines by the 
two lines h, k (see Fig. 44), which in their turn do not meet 
BC. [cf. SACCHERI, p. 42.] These lines, which the author calls 
parallels, have each a definite direction of parallelism. The 
line /5, of the figure, is the parallel to the right: k, to the left. 
The angle which the perpendicular AD makes with one of 
the parallels is the angle of parallelism for the length AD. 
LOBATSCHEWSKY uses the symbol TT (a) to denote the angle 
of parallelism corresponding to the length a. In the ordinary 
geometry, we have TT (a) = go always. In the geometry of 
LOBATSCHEWSKY, it is a definite function of a, tending to 
90 as a tends to zero, and to zero as a increases without 

From the definition of parallels the author then deduces 
their principal properties: 

88 IV. The Founders of Non-Euclidean Geometry (Contd.). 

That if AD is the parallel to BC for the point A, it is 
the parallel to BC in that direction for every point on AD 

That if AD is parallel to BC, then BC is parallel to 
AD [reciprocity] : 

That if the lines (2) and (3) are parallel to (i), then (2) 
and (3) are parallel to each other [transitivity] [cf. GAUSS, 
p. 72]; and that 

If AD and BC are parallel, AD is asymptotic to BC. 

Finally, the discussion of these questions is preceded by 
the theorems on the sum of the angles of a triangle, the 
same theorems as those already given by LEGENDRE, and 
still earlier by SACCHERI. There can be little doubt that Lo- 
BATSCHEWSKY was familiar with the work of LEGENDRE. 1 

But the most important part of the Imaginary Geometry 
is the construction of the formulae of trigonometry. 

To obtain these, the author introduces two new figures: 
the Horocycle [circle of infinite radius, cf. GAUSS, p. 74], and 
the Horosphere 2 [the sphere of infinite radius], which in the 
ordinary geometry are the straight line and plane, respect 
ively. Now on the Horosphere, which is made up of oo 2 
Horocycles, there exists a geometry analogous to the 
ordinary geometry, in which Horocycles take the place of 
straight lines. Thus LOBATSCHEWSKY obtains this first re 
markable result: 

The Euclidean Geometry [cf. WACHTER, p. 63], and, in 
particular, the ordinary plane trigonometry, hold upon the Hor 

i Cf. LOBATSCHEWSKY S criticism of LEGENDRE S attempt to 
obtain a proof of Euclid s Postulate in his New Principles of Geometry 
(ENGEL S translation, p. 68). 

* [LOBATSCHEWSKY uses the terms Grenzkreis, Grenzkugel in 
his German work: courbe-limite, horicycle, horisphere, surface-limite in 
his French work.] 

The Horocycle and Horocyclic Surface. 3Q 

This remarkable property and another relating to Co 
axal Horocycles [concentric circles with infinite radius] are 
employed by LOBATSCHEWSKY in deducing the formulae of 
the new Plane and Spherical Trigonometries x . The formulae 
of spherical trigonometry in the new system are found to be 
exactly the same as those of ordinary spherical trigonometry, 
when the elements of the triangle are measured in right- angles. 

41. It is well to note the form in which LOBATSCHEWSKY 
expresses these results. In the plane triangle ABC, let the 
sides be denoted by a, b, c, the angles by A, B, C; and let 
TT (a), TT (/), TT (c) be the angles of parallelism corresponding 
to the sides a t b, c. Then LOBATSCHEWSKY S fundamental 
formula is 

TT /7\ TT / \ s i n TT(^) sin TT (c) 

(4) cos A cos TT (J) cos TT (c) + - ^ (a) - ; - i. 

It is easy to see that this formula and that of TAURINUS 
[(0 P- 79] can be transformed into each other. 

To pass from that of TAURINUS to that of LOBATSCHEW 
SKY, we make use of (3) of p. 80, observing that the angle B, 
which appears in it, is TT (a). 

For the converse step, it is sufficient to use one of LO 
BATSCHEWSKY S results, namely : 

(5) *!!-.. 

This is the same as the equation (3) of TAURINUS, under 
another form. 

The constant a which appears in (5) is indeterminate. 
It represents the constant ratio of the arcs cut off two Coaxal 

1 It can be proved that the formulae of Non-Euclidean Plane 
Trigonometry can be obtained without the introduction of the 
Horosphere. The only result required is the relation between the 
arcs cut off" two Horocycles by two of their axes (cf. p. 90). Cf. 
H. LlEBMANN, Elementare Ableitung der nichteuklidischen Trigonometrie , 
Ber. d. kon. Sach. Ges. d. Wiss., Math. Phys. Klasse, (1907). 

GO IV. The Founders of Non-Euclidean Geometry (Contd.). 

Horocycles by a pair of axes, when the distance between 

these arcs is the unit of length. 

[Fig. 45-] 

If we choose, with LOBATSCHEW- 
SKY, a convenient unit, we are able 
to take a equal to e, the base of 
Natural Logarithms. If we wish, 
on the other hand, to bring Lo- 
BATSCHEWSKY S results into accord 

with the log. -spherical geometry of TAURINUS, or the Non-Eu- 

clidean geometry of GAUSS, we take 

a =<? . 
Then (5) becomes x 

,,. TT(*) ~T 

(5 ) tan ---- - = e 

2 > 

which is the same as 

This result at once transforms LOBATSCHEWSKY S equa 
tion (4) into the equation (i) of TAURINUS. 

It follows that: 

The log -spherical geometry of Taurinus is identical with 
the imaginary geometry [pangeometry] of Lobatschewsky. 

42. We add the most remarkable of the results which 
LOBATSCHEWSKY deduces from his formulae: 

(a) In the case of triangles whose sides are very small 
[infinitesimal] we can use the ordinary trigonometrical for 
mulae as the formulae of Imaginary Trigonometry, infinitesi 
mals of a higher order being neglected 1 . 

1 Conversely, the assumption that the Euclidean Geometry 
holds for the infinitesimally small can be taken as the starting 
point for the development of Non-Euclidean Geometry. It is one 
of the most interesting discoveries from the recent examination of 

Lobatschewsky s Trigonometry. QJ 

(b) If for a, b, c are substituted ia, ib, ic, the formulae 
of Imaginary Trigonometry are transformed into those of or 
dinary Spherical Trigonometry. 1 

(c) If we introduce a system of coordinates in two and 
three dimensions similar to the ordinary Cartesian coordinates, 
we can find the lengths of curves, the areas of surfaces, and 
the volumes of solids by the methods of analytical geometry. 

43. How was LOBATSCHEWSKY led to investigate the 
theory of parallels and to discover the Imaginary Geometry? 

We have already remarked that BARTELS, LOBATSCHEW 
SKY S teacher at Kasan, was a friend of GAUSS [p. 84]. If we 
now add that he and GAUSS were at Brunswick together dur 
ing the two years which preceded his call to Kasan [1807], 
and that later he kept up a correspondence with GAUSS, the 
hypothesis at once presents itself that they were not without 
their influence upon LOBATSCHEWSKY S work. 

We have also seen that before 1807 GAUSS had attempted 
to solve the problem of parallels, and that his efforts up till 
that date had not borne other fruit than the hope of overcom 
ing the obstacles to which his researches had led him. Thus 
anything that BARTELS could have learned from GAUSS before 
1807 would be of a negative character. As regards GAUSS S 

GAUSS S MSS. that the Princeps mathematiconim had already fol 
lowed this path. Cf. GAUSS, Werke, Bd. VIII, p. 255 264. 

Both the works of FLYE St. MARIE, [Theorie analytique sur la 
theorie ties paralleles, (Paris, 1871)], and of KILLING [Die nichteuklui- 
ischen Raumformen in analytischer Behandlung, (Leipzig, 1 881)], are 
founded upon this principle. In addition, the formulae of trigono 
metry have been obtained in a simple manner by the application 
of the same principle, and the use of a few fundamental ideas, by 
M. SIMON. [Cf. M. SIMON, Die Trigonometric in der absoluten Geometrie, 
CRELLE S Journal, Bd. 109, p. 187198 (1892)]. 

i This result justifies the method followed by TAURINUS in 
the construction of his log. -spherical geometry. 

O2 IV. The Founders of Non-Euclidean Geometry (Contd.). 

later views, it appears quite certain that BARTELS had no news 
of them, so that we can be sure that LOBATSCHEWSKY created 
his geometry quite independently of any influence from GAUSS.* 
Other influences might be mentioned: e. g., besides LEGENDRE, 
the works of SACCHERI and LAMBERT, which the Russian geo 
meter might have known, either directly or through KLUGEL 
and MONTUCLA. But we can come to no definite decision 
upon this question 2 . In any case, the failure of the demon 
strations of his predecessors, or the uselessness of his own 
earlier researches [1815 17], induced LOBATSCHEWSKY, as 
formerly GAUSS, to believe that the difficulties which had 
to be overcome were due to other causes than those to 
which until then they had been attributed. LOBATSCHEWSKY 
expresses this thought clearly in the New Principles of 
Geometry of 1825, where he says: 

The fruitlessness of the attempts made, since Euclid s 
time, for the space of 2000 years, aroused in me the suspicion 
that the truth, which it was desired to prove, was not contained 
in the data themselves; that to establish it the aid of experi 
ment would be needed, for example, of astronomical obser 
vations, as in the case of other laws of nature. When I had 
finally convinced myself of the justice of my conjecture and 
believed that I had completely solved this difficult question, 
I wrote, in 1826, a memoir on this subject {Exposition suc- 
cincte des principes de la Geomctrie\. * 

The words of LOBATSCHEWSKY afford evidence of a phil 
osophical conception of space, opposed to that of KANT, 
which was then generally accepted. The Kantian doctrine 
considered space as a subjective intuition, a necessary presup 
position of every experience. LOBATSCHEWSKY S doctrine was 

1 Cf. the work of F. ENGEL, quoted on p. .84. Zweiter Teil; 
Lobatschefskij s Leben und Schriften. Cap. VI, p. 373 383. 

2 Cf. SEGRE S work, quoted on p. 44. 

3 Cf. p. 67 of ENGEL S work named above. 

The Pangeometry. 93 

rather allied to sensualism and the current empiricism, and 
compelled geometry to take its place again among the ex 
perimental sciences. 1 

44. It now remains to describe the relation ofLoBAT- 
SCHEWSKY S Pangeometry to the debated question of the Eu 
clidean Postulate. This discussion, as we have seen, aimed 
at constructing the Theory of Parallels with the help of the 
first 28 propositions of Euclid. 

So far as regards this problem, LOBATSCHEWSKY, having 
defined parallelism, assigns to it the distinguishing features 
of reciprocity and transitivity. The property of equidistance 
then presents itself to LOBATSCHEWSKY in its true light. Far 
from being indissolubly bound up with the first 28 proposit 
ions of Euclid, it contains an element entirely new. 

The truth of this statement follows directly from the ex 
istence of the Pangeometry [a logical deductive science founded 
upon the said 28 propositions and on the negation of the 
Fifth Postulate], in which parallels are not equidistant, but are 
asymptotic. Further, we can be sure that the Pangeometry 
is a science in which the results follow logically one from the 
other, i. e., are free from internal contradictions. To prove 
this we need only consider, with LOBATSCHEWSKY, the analyt 
ical form in which it can be expressed. 

This point is put by LOBATSCHEWSKY toward the end of 
his work in the following way: 

Now that we have shown, in what precedes, the way in 
which the lengths of curves, and the surfaces and volumes of 
solids can be calculated, we are able to assert that the Pan- 
geometry is a complete system of geometry. A single glance 

1 Cf. The discourse on LOBATSCHEWSKY by A. VASILIEV, 
(Kasan, 1893). German translation by ENGEL in SciiLdMiLCii s Zeit- 
schrift, Bd. XI, p. 205 244 (1895). English translation by HALSTED, 
(Austin, Texas, 1895). 

Q4 IV. The Founders of Non-Euclidean Geometry (Contd.). 

at the equations which express the relations existing between 
the sides and angles of plane triangles, is sufficient to show 
that, setting out from them, Pangeometry becomes a branch of 
analysis, including and extending the analytical methods of 
ordinary geometry. We could begin the exposition of Pan- 
geometry with these equations. We could then attempt to 
substitute for these equations others which would express the 
relations between the sides and angles of every plane triangle. 
However, in this last case, it would be necessary to show 
that these new equations were in accord with the fundamental 
notions of geometry. The standard equations, having been 
deduced from these fundamental notions, must necessarily be 
in accord with them, and all the equations which we would 
substitute for them, if they cannot be deduced from the equa 
tions, would lead to results contradicting these notions. Our 
equations are, therefore, the foundation of the most general 
geometry, since they do not depend on the assumption that 
the sum of the angles of a plane triangle is equal to two right 
angles. x 

45. To obtain fuller knowledge of 
the nature of the constant k contained im- 
plicity in LOBATSCHEWSKY S formulae, and 
explicitly in those of TAURINUS, we must 
apply the new trigonometry to some actual 
case. To this end LOBATSCHEWSKY used a 
triangle ABC, in which the side BC (a) is 
equal to the radius of the earth s orbit, 
and A is a fixed star, whose direction is 
perpendicular to BC (Fig. 46). Denote 
by 2 p the maximum parallax of the star 
A. Then we have 


Fig. 46. 

1 Cf. the Italian translation of the Pangiomctrie, Giornale di 
Mat., T. V. p. 334; or p. 75 of the German translation referred to 
on p. 86. 

Astronomy and Lobatschewsky s Theory. 95 



t TIM >, 

But tan -- TT (a) = e~ k [cf. p. 90]. 

Therefore J 

But on the hypothesis / < , we have 

Also, tan zp = 2 tan/ 
I tan2/ 

= 2 (tan/ + tan^/ + tan 5 / + ...) 
Therefore we have 

T < tan 2 /- 

Take now, with LOBATSCHEWSKY, the parallax of Sirius 
as i", 24. 

From the value of tan 2/, we have 

<[ 0,000006012. 

This result does not allow us to assign a value to /, 
but it tells us that it is very great compared with the diam 
eter of the earth s orbit. We could repeat the calculation 
for much smaller parallaxes, for example o",i, and we 
would find k to be greater than a million times the diameter 
of the earth s orbit. 

Thus, if the Euclidean Geometry and the Fifth Postul 
ate are to hold in actual space, k must be infinitely great. 
That is to say, there must be stars whose parallaxes are in 
definitely small. 

However it is evident that we can never state whether 
this is the case or not, since astronomical observations will 

C)6 IV. The Founders of Non-Euclidean Geometry (Contd.)- 

always be true only within certain limits. Yet, knowing the 
enormous size of k in comparison with measurable lengths, 
we must, with LOBATSCHEWSKY, admit that the Euclidean 
hypothesis is valid for all practical purposes. 

We would reach the same conclusion if we regarded 
the question from the standpoint of the sum of the angles of 
a triangle. The results of astronomical observations show that 
the defect of a triangle, whose sides approach the distance 
of the earth from the sun, cannot be more than o",ooc>3. 
Let us now consider, instead of an astronomical triangle, one 
drawn on the Earth s surface, the angles of which can be 
directly measured. In consequence of the fundamental theorem 
that the area of a triangle is proportional to its defect, the 
possible defect would fall within the limits of experimental 
error. Thus we can regard the defect as zero in experimental 
work, and Euclid s Postulate will hold in the domain of ex 
perience. 1 

Johann Bolyai [1802 1860]. 

46. J. BOLYAI a Hungarian officer in the Austrian 
army, and son of WOLFGANG BOLYAI, shares with LOBAT 
SCHEWSKY the honour of the discovery of Non-Euclidean geo 
metry. From boyhood he showed a remarkable aptitude for 
mathematics, in which his father himself instructed him. The 
teaching of WOLFGANG quickly drew JOHANN S attention to 
Axiom XL To its demonstration he set himself, in spite of 
the advice of his father, who sought to dissuade him from 
the attempt. In this way the theory of parallels formed the 
favourite occupation of the young mathematician, during his 
course [1817 22] in the Royal College for Engineers at 

i For the contents of this section, cf. LOBATSCHEWSKY, On 
the Principles of Geometry. See p. 22 24 of ENGEL s work named 
on p. 84. Also ENGEL S remarks on p. 248252 of the same work. 

Johann Bolyai s Earlier Work. 


At this time JOHANN was an intimate friend of CARL 
SZASZ [i 798-185 3] and the seeds of some of the ideas, which 
led BOLYAI to create the Absolute Science of Space, were sown 
in the conversations of the two eager students. 

It appears that to SZASZ is due the distinct idea of con 
sidering the parallel through B to the line AM as the limit 
ing position of a secant BC turning in a definite direction 
about B\ that is, the idea of consid 
ering BC as parallel to AM, when 
BC, in the language of SZASZ, de 
taches itself (springs away) from AM 
(Fig. 47). To this parallel BOLYAI 
gave the name of asymptotic parallel 
or asymptote, [cf. SACCHERI]. From 
the conversations of the two friends 
were also derived the conception of 
the line equidistant from a straight line, 
and the other most important idea of 
the Paracycle (limiting curve or horo- 
cycle of LOBATSCHEWSKY). Further they 
recognised that the proof of Axiom XI would be obtained 
if it could be shown that the Paracycle is a straight line. 

When SZASZ left Vienna in the beginning of 1821 to 
undertake the teaching of Law at the College of Nagy-Enyed 
(Hungary), JOHANN remained to carry on his speculations 
alone. Up till 1820 he was filled with the idea of finding 
a proof of Axiom XI, following a path similar to that of 
SACCHERI and LAMBERT. Indeed his correspondence with 
his father shows that he thought he had been successful in 
his aim. 

The recognition of the mistakes he had made was the 
cause of JOHANN S decisive step towards his future discoveries, 
since he realised that one must do no violence to nature, 
nor model it in conformity to any blindly formed chimaera; 


^8 Iv - The Founders of Non-Euclidean Geometry (Contd.) 

that, on the other hand, one must reguard nature reasonably 
and naturally, as one would the truth, and be contented only 
with a representation of it which errs to the smallest possible 

JOHANN BOLYAI, then, set himself to construct an abso 
lute theory of space, following the classical methods of the 
Greeks: that is, keeping the deductive method, but without 
deciding a priori on the truth or error of the FifthPostulate. 

47. As early as 1823 BOLYAI had grasped the real 
nature of his problem. His later additions only concerned 
the material and its formal expression. At that date he had 
discovered the formula: 

e k = tan ^- ; , 

connecting the angle of parallelism with the line to which it 
corresponds [cf. LOBATSCHEWSKY, p. 89]. This equation is 
the key to all Non-Euclidean Trigonometry. To illustrate the 
discoveries which JOHANN made in this period, we quote the 
following extract from a letter which he wrote from Temesvar 
to his father, on Nov. 3, 1823: I have now resolved to pub 
lish a work on the theory of parallels, as soon as I shall have 
put the material in order, and my circumstances allow it. I 
have not yet completed this work, but the road which I have 
followed has made it almost certain that the goal will be 
attained, if that is at all possible: the goal is not yet reached, 
but I have made such wonderful discoveries that I have been 
almost overwhelmed by them, and it would be the cause of 
continual regret if they were lost. When you will see them, 
you too will recognize it. In the meantime I can say only 
this : / have created a new universe from nothing. All that I 
have sent (you till now is but a house of cards compared to 
the tower. I am as fully persuaded that it will bring me 
honour, as if I had already completed the discovery. 

J. Bolyai s Theory of Parallels. o/} 

WOLFGANG expressed the wish at once to add his son s 
theory to the Tentamen since if you have really succeeded 
in the question, it is right that no time be lost in making it 
public, for two reasons: first, because ideas pass easily from 
one to another, who can anticipate its publication; and se 
condly, there is some truth in this, that many things have an 
epoch, in which they are found at the same time in several 
places, just as the violets appear on every side in spring. 
Also every scientific struggle is just a serious war, in which 
I cannot say when peace will arrive. Thus we ought to 
conquer when we are able, since the advantage is always to 
the first comer. 

Little did WOLFGANG BOLYAI think that his presentiment 
would correspond to an actual fact (that is, to the simulta 
neous discovery of Non-Euclidean Geometry by the work of 

In 1825 JOHANN sent an abstract of his work, among 
others, to his father and to J.WALTER VON ECKWEHR [1789 
1857], his old Professor at the Military School. Also in 1829 
he sent his manuscript to his father. WOLFGANG was not 
completely satisfied with it, chiefly because he could not see 
why an indeterminate constant should enter into JOHANN S 
formulae. None the less father and son were agreed in 
publishing the new theory of space as an appendix to the 
first volume of the Tentamen . 

The title of JOHANN BOLYAI S work is as follows. 

Appendix scientiam spatii absolute veram exhibens: a 
7 eritatf aut falsitate Axiomatis XL Euclidei, a priori hand 
unquam dccidenda, independentem : adject a ad casum falsitatis 
quadratura circuit geometrical 

1 A reprint Edition de Luxe was issued by the Hungarian 
Academy of Sciences, on the occasion of the first centenary of 
the birth of the author (Budapest, 1902). See also the English 


IOO IV. Tne Founders of Non-Euclidean Geometry (Contd.). 

The Appendix was sent for the first time [June, 1831] 
to GAUSS, but did not reach its destination; and a second 
time, in January, 1832. Seven weeks later (March 6, 1832), 
GAUSS replied to WOLFGANG thus: 

"If I commenced by saying that I am unable to praise 
this work (by JOHANN), you would certainly be surprised 
for a moment. But I cannot say otherwise. To praise it, 
would be to praise myself. Indeed the whole contents of 
the work, the path taken by your son, the results to which he 
is led, coincide almost entirely with my meditations, which 
have occupied my mind partly for the last thirty or thirty- 
five years. So I remained quite stupefied. So far as my 
own work is concerned, of which up till now 1 have put little 
on paper, my intention was not to let it be published during 
my lifetime. Indeed the majority of people have not clear 
ideas upon the questions of which we are speaking, and I 
have found very few people who could regard with any special 
interest what I communicated to them on this subject. To 
be able to take such an interest it is first of all necessary 
to have devoted careful thought to the real nature of what is 
wanted and upon this matter almost all are most uncertain. 
On the other hand it was my idea to write down all this later 
so that at least it should not perish with me. It is therefore a 
pleasant surprise for me that I am spared this trouble, and I 
am very glad that it is just the son of my old friend, who 
takes the precedence of me in such a remarkable manner." 

WOLFGANG communicated this letter to his son, adding: 
"GAUSS S answer with regard to your work is very satis- 

translation by HALSTED, The Science Absolute of Space, (Austin, Texas^ 
1896). An Italian translation by G. B. BATTAGLINI appeared in the 
Giornale di Mat., T. VI, p. 97 115 (1868). Also a French trans 
lation by HOUEL, in Mem. de la Soc. des Sc. de Bordeaux, T. V- 
p. 189248 (1867). Cf. also FRISCHAUF, Absolute Geometrie nach 
Johann Bolyai, (Leipzig, Teubner, 1872). 

Gauss s Praise of Bolyai s Work. IOI 

factory and redounds to the honour of our country and of 
our nation." 

Altogether different was the effect GAUSS S letter pro 
duced on JOHANN. He was both unable and unwilling to 
convince himself that others, earlier than and independent of 
him, had arrived at the Non-Euclidean Geometry. Further he 
suspected that his father had communicated his discoveries 
to GAUSS before sending him the Appendix and that the latter 
wished to claim for himself the priority of the discovery. 
And although later he had to let himself be convinced that 
such a suspicion was unfounded, JOHANN always regarded 
the "Prince of Geometers" with an unjustifiable aversion. 1 

48. We now give a short description of the most 
important results contained in JOHANN BOLYAI S work: 

a) The definition of parallels and their properties in 
dependent of the Euclidean postulate. 

b) The circle and sphere of infinite radius. The geo 
metry on the sphere of infinite radius is identical with ordi 
nary plane geometry. 

c) Spherical Trigonometry is independent of Euclid s 
Postulate. Direct demonstration of the formulae. 

d) Plane Trigonometry in Non-Euclidean Geometry. 
Applications to the calculation of areas and volumes. 

e) Problems which can be solved by elementary me 
thods. Squaring the circle, on the hypothesis that the Fifth 
Postulate is false. 

While LOBATSCHEWSKY has given the Imaginary Geo 
metry a fuller development especially on its analytical side, 

i For the contents of this and the preceding article seeSxACKEL, 
Die Entdeckung der nichteuklidischen Geometric durch Johann Bolyai, 
Math. u. Naturw. Berichte aus Ungarn. Bd. XVII, [1901]. 

Also SxAcKEL u. ENGEL. Gauss, die beiden Bolyai und die 
nichteuklidische Geometrie. Math. Ann. Bd. XLIX, p. 149167 [1897], 
Bull. Sc. Math. (2) T. XXI, pp. 206228 [1897]. 

IO2 IV. The Founders of Non-Euclidean Geometry (Contd.). 

BOLYAI entered more fully into the question of the depen 
dence or independence of the theorems of geometry upon 
Euclid s Postulate. Also while LOBATSCHEWKY chiefly sought 
to construct a system of geometry on the negation of the 
said postulate, JOHANN BOLYAI brought to light the pro 
positions and constructions in ordinary geometry which are 
independent of it. Such propositions, which he calls ab 
solutely true, pertain to the absolute science of space. We 
could find the propositions of this science by comparing 
Euclid s Geometry with that of LOBATSCHEWSKY. Whatever 
they have in common, e. g. the formulae of Spherical Trigon 
ometry, pertains to the Absolute Geometry. JOHANN BOLYAI, 
however, does not follow this path. He shows directly, that 
is independently of the Euclidean Postulate, that his propos 
itions are absolutely true. 

49. One of BOLYAI S absolute theorems, remarkable 
for its simplicity and neatness, is the following: 

The sines of the angles of a rectilinear triangle are to one 
another as the circumferences of the circles whose radii are 

equal to the opposite sides. 


B _ 

B M 

Fig. 48- 

Let ABC be a triangle in which C is a right angle, and 
BB the perpendicular through B to the plane of the triangle. 

Draw the parallels through A and C to BB in the 
same sense. 

Then let the Horosphere be drawn through A (eventually 
the plane) cutting the lines AA\ BB and CC , respectively, 
in the points A, M, and N. 

Bolyai s Theorem. 103 

If we denote by a , b\ c the sides of the rectangular 
triangle AMN on the Horosphere, it follows from what has 
been said above [cf. g 48 (b)] that 

sin AMN = . 


But two arcs of Horocycles on the Horosphere are pro 
portional to the circumferences of the circles which have 
these arcs for their (horocyclic) radii. 

If we denote by circumf. x the circumference of the 
circle whose (horocyclic) radius is x , we can write: 

A *r*T circumf. b 
sm AMN = -. -- - ( -,. 
circumf. c 

On the other hand, the circle traced on the Horosphere 
with horocyclic radius of length # , can be regarded as the 
circumference of an odinary circle whose radius (rectilinear) 
is half of the chord of the arc 2 x of the Horocycle. 

Denoting by Q x the circumference of the circle whose 
(rectilinear) radius is x, and observing that the angles ABC 
and AMN are equal, the preceding equation taken from 

From the property of the right angled triangle ABC 
expressed by this equation, we can deduce BOLYAI S theorem 
enunciated above, just as from the Euclidean equation 

sin ABC - y 

we can deduce that the sines of the angles of a triangle are 
proportional to the opposite sides. [Appendix 25.] 

BOLYAI S Theorem may be put shortly thus: 
(0 O : O : O = sin A : sin B : sin C. 

If we wish to discuss the geometrical systems separately 
we will have 

(i) In the case of the Euclidean Hypothesis, 

IO4 Iv The Founders of Non-Euclidean Geometry (Contd.). 

Thus, substituting in (i), we have 
( i ) a : b : c : = sin A : sin B : sin C. 

(ii) In the case of the Non-Euclidean Hypothesis, 

f = TU k 


sinh - 

Then substituting in (i) we have 

(i ") sinh : sinh : sinh -r = sin A : sin B : sin C. 

K K K 

This last relation may be called the Sine Theorem of the 
Bolyai-Lobatschewsky Geometry. 

From the formula (i) BOLYAI deduces, in much the 
same way as the usual relations are obtained from (i ) t the 
proportionality of the sines of the angles and the opposite sides 
in a spherical triangle. From this it follows that Spherical 
Trigonometry is independent of the Euclidean Postulate 
{Appendix S 26]. 

This fact makes the importance of BOLYAI S Theorem 
still clearer. 

50. The following construction for a parallel through 
the point Z> to the straight line AN belongs also to the Ab 
solute Geometry [Appendix S 34]. 

Draw the perpendiculars DB and AE to AN [Fig. 49]. 

Fig. 49. 

Also the perpendicular DE to the line AE. The angle 
EDB of the quadrilateral ABDE, in which three angles 

Bolyai s Parallel Construction. 105 

are right angles, is a right angle or an acute angle, according 
as ED is equal to or greater than AB. 

With centre A describe a circle whose radius is equal 
to ED. 

It will intersect DB at a point O, coincident with B or 
situated between B and D. 

The angle which the line AO makes with DB is the 
angle of parallelism corresponding to the segment BD? 
{Appendix 27.] 

Therefore a parallel to AN through D can be con 
structed by drawing the line DM so that <$ BDM is 
equal to -^ AOB. 2 

i We give a sketch of BOLYAI S proof of this theorem: The 
circumferences of the circles with radii AB and ED, traced out 
by the points B and D in their rotation about the line AE, can 
be considered as belonging, the first to the plane through A per 
pendicular to the axis AE, the second to an Equidistant Surface 
for this plane. The constant distance between the surface and 
the plane is the segment BD = d. The ratio between these two 
circumferences is thus a function of d only. Using BOLYAI S 
Theorem, $ 49, and applying it to the two rightangled triangles 
ADE and ADB, this ratio can be expressed as 
QA:Q ED = sin u : sin v. 

From this it is clear that the ratio sin u : sin v does not vary if 
the line AE changes its position, remaining always perpendicular 
to AB, while d remains fixed. In particular, if the foot of AE 
tends to infinity along AN, u tends to TT (d) and v to a right angle. 

On the other hand in the right-angled triangle AOB, we have 
the equation 

O AB : O AO = sin AOB : I. 

This, with the preceding equation, is sufficient to establish the 
equality of the angles TT (d) und AOB. 
2 Cf. Appendix III to this volume. 


The Founders of Non-Euclidean Geometry (Contd.). 

51. The most interesting of the Non-Euclidean con 
structions given by BOLYAI is that for the squaring of the 
circle. Without keeping strictly to BOLYAI S method, we shall 
explain the principal features of his construction. 

But we first insert the converse of the construction of 
50, which is necessary for our purpose. 

On the Non-Euclidean Hypothesis to draw the segment 
which corresponds to a given (acute) angle of parallelism. 

Assuming that the theorem, that the three perpendiculars 
from the angular points of a triangle on the opposite sides 
intersect eventually, is also true in the Geometry of BOLYAI- 
LOBATSCHEWSKY, on the line AB bounding the acute angle 
BAA take a point B, such that the parallel B& to A A 
through B makes an acute angle (ABB ) with AB. [Fig. 50.] 




Fig. 50. 

The two rays AA\ BB , and the line AB may be 
regarded as the three sides of a triangle of which one angular 
point is CQO , common to the two parallels AA , BB . Then 
the perpendiculars from A, B, to the opposite sides, meet in 
he point O inside the triangle, and the perpendicular from 
CQO to AB also passes through O. 

Thus, if the perpendicular OL is drawn from O to AB, 
the segment AL will have been found which corresponds to 
the angle of parallelism BAA . 

Bolyai s Parallel Construction (Contd.). 


As a particular case the angle BAA could be 45. 
Then AL would be Schweikart s Constant [cf. p. 76]. 

We note that the problem which we have just solved 
could be enunciated thus: 

To draw a line which shall be parallel to one of the lines 
bounding an acute angle and perpendicular to the other* 

52. We now show how the preceding result is used 
to construct a square equal in area to the maximum triangle. 

The area of a triangle being 

2 (TT ^A ^B < C), 

the maximum triangle, i. e. that for which the three angular 
points are at infinity, will have for area 

A = 2 TT. 

To find the angle uu of a square whose area is k 2 Ti, we need 
only remember (LAMBERT, p. 46) that the area of a polygon, 
as well as of a triangle, is proportional to its defect. Thus 
we have the equation 

k* TT = k 2 (2 TT 4 Uj), 

from which it follows that 


4-W-45 - 

Assuming this, let us consider the 
right-angled triangle OAM (Fig. 51), 
which is the eighth part of the required 
square. Putting OM = #, and ap 
plying the formula (2) of p. 80 we 




or cosh 

cos 22 30 
sin 450 

sin 67 30 
sin 450 


1 BOLYAI S solution {Appendix., S 35] is, however, more compli- 

IO8 IV- The Founders of Non-Euclidean Geometry (Contd.). 

If we now draw, as in 51, the two segments b , c , 
which correspond to the angles 67 30 and 45, and if we 
remember that [cf. p. 90 (6)] 

, * i 

COSh -r- = -r-rrT-r, 
k smTT(jr) 

the following relation must hold between a, b and c > 
cosh -r cosh -r = cosh -r-. 

K K K 

Finally if we take b as side, and c as hypotenuse of a right- 
angled triangle, the other side of this triangle, by formula (i) 
of p. 79, is determined by the equation 

cosh -v cosh -.- = cosh -T-, 

K K K 

Then comparing these two questions, we obtain 

a = a. 

Constructing a in this way, we can immediately find the 
square whose area is equal to that of the maximum triangle. 

53. To construct a circle whose area shall be equal 
to that of this square, that is, to the area of the maximum 
triangle, we must transform the expression for the area of 
a circle of radius r 

2 TT k 2 ( cosh -- i ) , 

given on p. 8 1, by the introduction of the angle of parallelism 
TT( J, corresponding to half the radius. 

Then we have J for the area of this circle 

On the other hand if the two parallels A A and BB 
are drawn from the ends of the segment AB^ making equal 
angles with AB, we have 

Using the result tan n ^ = e T x k 

The Square of area TT/ >2 . 


AAB <r B BA = TT (~ \ 
where AB = r [Fig. 52]. 

Now draw ^C, perpendicular to BB , and ^4Z> perpen 
dicular to ^C; also put 

<: C/l# a, < ZX4^ = 2. 
Then we have 

cot IT j cot a + i 

cot a cot TT ( 

tan z 

It is easy to eliminate a from this last result by means 
of the trigonometrical formulae for the triangle ABC and so 

tan z = 

Substituting this in the expression found for the area of 
the circle, we obtain for that area 

TT k 2 tan 2 z. 

This formula, proved in an- D A B 

other way by BOLYAI [Appendix 
8 43], allows us to associate a 
definite angle z with every circle. 
If z were equal to 45, then we \ z \ 
would have 

for the area of the correspond 
ing circle. 

1 Indeed, in the rightangled triangle ABC, we have cot TT ( ) 

= cosh -_, From this, since cosh -. = 2 si 

* k 

2 TT ( 2 J + i we deduce, first, that 

+ I = 

I IO IV. The Founders of Non-Euclidean Geometry (Contd.). 

That is : the area of the circle, for which the angle z is 
>, is equal to the area of the maximum triangle, and thus 
to that of the square of 52. 

If z = *^A AD (Fig. 51) is given, we can find r by 
the following construction: 

(i) Draw the line AC perpendicular to AD. 

(ii) Draw BB parallel to AA and perpendicular to 

(iii) Draw the bisector of the strip between AA and 
BB . 

[By the theorem on the concurrency of the bisectors of 
the angles of a triangle with an infinite vertex.] 

(iv) Draw the perpendicular AB to this bisector. The 
segment AB bounded by AA and BB is the required 
radius r. 

54. The problem of constructing a polygon equal to 
a circle of area TT k 2 tan 2 z is, as BOLYAI remarked, closely 
allied with the numerical value of tan z. It is resolvable 
for every integral value of tan 2 z, and for every fractional 
value, provided that the denominator of the fraction, re 
duced to its lowest terms, is included in the form assigned by 
GAUSS for the construction of regular polygons [Appendix 


The possibility of constructing a square equal to a 
circle leads JOHANN to the conclusion "habeturque aut Axi- 
oma XI Euclidis verum, aut quadratura circuit geometrica;. 

cot TT ( ) cot a = 2 cot2 TT ( 
and next that 

cot a cot TT^-jJ = M + tan2 TT ( 3 )) cot TT (y 

These equations allow the expression for tan z to be written down 
in the required form. 

The Quadrature of the Circle. Ill 

etsi hucusque indecisum manserit, quodnam ex his duobus 
revera locum habeat." 

This dilemma seemed to him at that time [1831] im 
possible of solution, since he closed his work with these 
words: "Superesset denique (ut res omninumero absolvatur), 
impossibilitatem (absque suppositione aliqua) decidenda, 
num X (the Euclidean system) aut aliquod (et quodam) SJthe 
Non-Euclidean system) sit, demonstrare : quod tamen occasi 
on! magis idoneae reservatur." 

JOHANN, however, never published any demonstration 
of this kind. 

55. After 1831 BOLYAI continued his labours at his 
geometry, and in particular at the following problems: 

1. The connection between Spherical Trigonometry and 
Non-Euclidean Trigonometry. 

2. Can one prove rigorously that EUCLID S Axiom is 
not a consequence of what precedes it ? 

3. The volume of a tetrahedron in Non-Euclidean geo 

As regards the first of these problems, beyond estab 
lishing the analytical relation connecting the two trigono 
metries fcf. LOBATSCHEWSKY, p. 90], BOLYAI recognized that 
in the Non-Euclidean hypothesis there exist three classes of 
Uniform Surfaces 1 on which the Non-Euclidean trigono 
metry, the ordinary trigonometry, and spherical trigonometry 
respectively hold. To the first class belong planes and hyper- 
spheres [surfaces equidistant from *a plane]; to the second, 
the paraspheres [LOBATSCHEWSKY S Horospheres]; to the 
third, spheres. The paraspheres are the limiting case 
when we pass from the hyperspherical surfaces to the 
spherical. This passage is shown analytically by making a 

1 BOLYAI seems to indicate by this name the surfaces which 
behave as planes, with respect to displacement upon themselves. 

112 IV. The Founders of Non-Euclidean Geometry (Contd.). 

certain parameter, which appears in the formulae, vary con 
tinuously from the real domain to the purely imaginary 
through infinity [cf. TAURINUS, p. 82]. 

As to the second problem, that regarding the impos 
sibility of demonstrating Axiom XI, BOLYAI neither succeeded 
in solving it, nor in forming any definite opinion upon it. 
For some time he believed that we could not, in any way, 
decide which was true, the Euclidean hypothesis or the 
Non-Euclidean. Like LOBATSCHEWSKY, he relied upon the 
analytical possibility of the new trigonometry. Then we find 
JOHANN returning again to the old ideas, and attempting a 
new demonstration of Axiom XL In this attempt he applies 
the Non-Euclidean formulas to a system of five coplanar 
points. There must necessarily be some relation between 
the distance of these points. Owing to a mistake in his 
calculations JOHANN did not find this relation, and for some 
time he believed that he had proved, in this way, the false 
hood of the Non-Euclidean hypothesis and the absolute truth 
of Axiom Xf. 1 

However he discovered his mistake later, but he did 
not carry out further investigations in this direction, as the 
method, when applied to six or more points, would have in 
volved too complicated calculations. 

The third of the problems mentioned above, that re 
garding the tetrahedron, is of a purely geometrical nature. 
BOLYAI S solutions have been recently discovered and pub- 

1 The title of the paper which contains JOHANN S demon 
stration is as follows: "Beweis dcs bis nun auf der Erde immer 
noch zweifelhaft gewesenen, tveltberiihmten und y aJs der gesammten 
Kaum- und Bewegungslehre zu Grunae dienend, auch in der That 
allerhochstwichtigsten //. Euclid schen Axioms von J. Bolyai von Bolya, 
k. fc. Genie-Stabshauptmann in Pension. Cf. STACKEL s paper: Unter~ 
suchungcn aus der Absoluten Geometric mis Johann Bolyais Nachlafi. 
Math. u. Naturw. Berichte aus Ungarn. Bd. XVIII, p. 280307 (1902). 
We are indebted to this paper for this section S 55- 

Bolyai s Later Work. j j ? 

lished by STACKEL [cf. p. 112 note i], LOBATSCHEWSKY 
had been often occupied with the same problem from 1829 , 
and GAUSS proposed it to JOHANN in his letter quoted on 
p. 100. 

Finally we add that J. BOLYAI heard of LOBATSCHEWSKY S 
Geometrische Untersuchungen in 1848: that he made them 
the object of critical study 2 : and that he set himself to com 
pose an important work on the reform of the Principles of 
Mathematics with the hope of prevailing over the Russian. 
He had planned this work at the time of the publication of 
the Appendix, but he never succeeded in bringing it to a 

The Absolute Trigonometry. 

56. Although the formulae of Non-Euclidean trigono 
metry contain the ordinary relations between the sides and 
angles of a triangle as a limiting case [cf. p. 80], yet they do 
not form a part of what JOHANN BOLYAI called Absolute Geo 
metry. Indeed the formulae do not apply at once to the two 
classes of geometry, and they were deduced on the suppos 
ition of the validity of the Hypothesis of the Acute Angle. 
Equations directly applicable both to the Euclidean case and 
to the Non-Euclidean case were met by us in 49 and they 
make up BOLYAI S Theorem. They are three in number, only 
two of them being independent. Thus they furnish a first 
set of formulae of Absolute Trigonometry. 

x Cf. p. 53 et seq., of the work quoted on p. 84. Also 
LIEBMANN S translation, referred to in Note 2, p. 85. 

2 Cf. P. STACKEL und J. KURSCHA K: Johann Bolyais Be- 
merkungen itber N. Lobalschefskijs Geometrische UntersucJiungen zitr 
Jlieorie der Parallellinicn, Math. u. Naturw. Berichte aus Ungarn, 
Bd. XVIII, p. 250279 (1902). 

5 Cf. P. SxACKEL: Johann Bolyais Raumlehre, Math. u. Naturw. 
Berichte aus Ungarn, Bd. XIX (1903). 


HA IV. The Founders of Non-Euclidean Geometry (Contd.). 

Other formulae of Absolute Trigonometry were given in 
1870 by the Belgian geometer, DE TILLY, in his Etudes de 
Mecanique Abstraite. x 

The formulae given by DE TILLY refer to rectilinear tri 
angles, and were deduced by means of kinematical con 
siderations, requiring only those properties of a bounded 
region of a plane area, which are independent of the value 
of the sum of the angles of a triangle. 

In addition to the function O^* which we have already 
met in BOLYAI S formulae, those of DE TILLY contain another 
function Ex defined in the following way: 

Let r be a straight line and / the equidistant curre, 
distant x from r. Since the arcs of / are proportional to their 
projections on r, it is clear that the ratio between a (recti 
fied) arc of / and its projection does not depend on the 
length of the arc, but only on its distance x from r. DE 
TILLY S function Ex is the function which expresses this ratio. 

On this understanding, the Formulae of Absolute Trigon 
ometry for the right-angled triangle ABC are as follows : 
B (i) JQ0 = Qc sin A 

JO b = O f sin B 

( 2 ) jcos A = Ea. sin B 
[cos B = Eb. sin A 

(3) EC = Ea. Eb. 

The set (i) is equivalent to BOLYAI S 
Theorem for the Right- Angled Triangle. 
All the formulae of Absolute Trigono 
metry could be derived by suitable com 
bination of these three sets. In particular, for the right-angled 
triangle, we obtain the following equation: 

1 Me"moires couronns et autres Me"moires, Acad. royale de 
Belgique. T. XXI (1870). Cf. also the work by the same author: 
Essai sur les principes fondamentaux de la ?eonittrie et dc la Mecanique, 
Mem. de la Soc. des Sc. de Bordeaux. T. Ill (cah. I) (1878). 

The Absolute Trigonometry. Ijr 

O 2 * (Ea + Eb. EC} -f O 2 ^- (Eb + EC. Ea) 

= O 2 (-& + -* Eb). 

This can be regarded as equivalent to the Theorem of 
Pythagoras in the Absolute Geometry. 1 

57. Let us now see how we can deduce the results 
of the Euclidean and Non-Euclidean geometries from the 
equations of the preceding article. 

Euclidean Case. 

The Equidistant Curve (I) is a straight line [that is, Ex 
= i], and the perimeters of circles are proportional to 
their radii. 

Thus the equations (i) become 
(i ) (a = c sin A 

\b = c sin B. 

The equations (2) give 
(2 ) cos A = sin B> cos B = sin A. 

Therefore A + B = 90. 

Finally the equation (3) reduces to an identity. 

The equations (V) and (2) include the whole of ordin 
ary trigonometry. 

Non-Euclidean Case. 

Combining the equations (i) and (2) we obtain 

If we now apply the first of equations (2) to a right- 
angled triangle whose vertex A goes off to infinity, so that 
the angle A tends to zero, we shall have 

Lt cos A = Lt (Ea. sin B). 

But Ea is independent of A-, also the angle B, in the 
limit, becomes the angle of parallelism corresponding to a, 
i. e. TT (a). 

1 Cf. R. BONOLA, La trigonometria assjluia sscondo Giovann, 
Bolyai. Rend. Istituto Lombardo (2). T. XXXVIII (19OS\ 


Il6 IV- The Founders of Non-Euclidean Geometry (Contd.). 
Therefore we have 

A similar result holds for Eb. 

Substituting these in equation (5) we obtain 

Q 2 a Q 2 b 

cot2TT(a) " " cot 2 !!^) 
from which 

O = Q 

cot IT (a) cot at (l>) 

This result, with the expression for Ex, allows us at 
once to obtain from the equations (i), (2), (3), the formulae 
of the Trigonometry of BOLYAI-LOBATSCHEWSKY : 
fcot TT (a) = cot TT (c) sin A 
(cot TT (b) = cot TT (c) sin B, 
[sin A = cos B sin TT (b) 
[sin B = cos A sin TT (a), 
(3") sin TT (c) = sin TT (a) sin TT (b). 

These relations between the elements of every right- 
angled triangle were given in this form by LoBATSCHEWSKY. 1 
If we wish to introduce direct functions of the sides, instead 
of the angles of parallelism TT (a\ TT (b) and TT (c) t it is 
sufficient to remember [p. 90] that 

tan ^ = <-*/*. 

We can thus express the circular functions of TT (x) in 
terms of the hyperbolic functions of x. In this way the pre 
ceding equations are replaced by the following relations: 

, , , a . , c 

(i ) smh -T- = sinn -T- sin A 

sinh -=- = sinh sin 

k /: 

i Cf. e. g., The Geometrisehe Untersuchungen of LOBATSCHEWSKV 
referred to on p. 86. 

Absolute Trigonometry and Spherical Trigonometry. \\*j 

(2" ) cos A = sin B cosh -| 

cos B = sin A cosh , , 
(3 ") cosh ~ = cosh - 

58. The following remark upon Absolute Trigono 
metry is most important: If we regard the elements in its 
formulae as elements of a spherical triangle, we obtain a system 
of equations which hold also for Spherical Triangles. 

This property of Absolute Trigonometry is due to the 
fact, already noticed on p. 114, that it was obtained by the 
aid of equations which hold only for a limited region of the 
plane. Further these do not depend on the hypothesis of the 
angles of a triangle, so that they are valid also on the sphere. 

If it is desired to obtain the result directly, it is only 
necessary to note the following facts: 

(i) In Spherical Trigonometry the circumferences of 
circles are proportional to the sines of their (spherical) radii, 
so that the first formula for right-angled spherical triangles 

sin a sin c sin A 
is transformed at once into the first of the equations (i). 

(ii) A circle of (spherical) radius b can be [con 
sidered as a curve equidistant from the concentric great 
circle, and the ratio Eb for these two circles is given by 

cos b. 


Thus the formulae for right-angled spherical triangles 
cos A = sin B cos a, 
cos ^ = cos a cos , 

1 1 8 IV. The Founders of Non-Euclidean Geometry (Contd.). 

are transformed immediately into the equations (2) and (3) 
by means of this result. 

Thus the formulae of Absolute Trigonometry also hold 
on the sphere. 

Hypotheses equivalent to Euclid s Postulate. 

59. Before leaving the elementary part of the sub 
ject, it seems right to call the attention of the reader to the 
position occupied in the general system of geometry by certain 
propositions, which are in a certain sense hypotheses equivalent 
to the Fifth Postulate. 

That our argument may be properly understood, we 
begin by explaining the meaning of this equivalence. 

Two hypotheses are absolutely equivalent when each of 
them can be deduced from the other without the help of any 
new hypothesis. In this sense the two following hypotheses 
are absolutely equivalent: 

a) Two straight lines parallel to a third are parallel to 
each other; 

b) Through a point outside a straight line one and only 
one parallel to it can be drawn. 

This kind of equivalence has not much interest, since 
the two hypotheses are simply two different forms of the 
same proposition. However we must consider in what way 
the idea of equivalence can be generalised. 

Let us suppose that a deductive science is founded 
upon a certain set of hypotheses, which we will denote by 
{A, B, C\... H}. Let M and uWbe two new hypotheses such 
that N can be deduced from the set { A, B, C . . . H, M], 
and M from the set {A, , C . . . H, N\ 

We indicate this by writing 

Absolute and Relative Equivalence. ng 



We shall now extend the idea of equivalence and say that 
the two hypotheses J/, N are equivalent relatively to the 
fundamental set {A, J3, C . . . H}. 

It has to be noted that the fundamental set {A, , C 
. . . H} has an important place in this definition. Indeed it 
might happen that by diminishing this fundamental set, leav 
ing aside, for example, the hypothesis A, the two deductions 

{JB, C,...H,M} .). N 


could not hold simultaneously. 

In this case the hypotheses M, N are not equivalent 
with respect to the new fundamental set \B, C . . . J?}. 

After these explanations of a logical kind, let us see 
what follows from the discussion in the preceding chapters 
as to the equivalence between such hypotheses and the 
Euclidean hypothesis. 

We assume in the first place as fundamental set of 
hypotheses that formed by the postulates of Association (A), 
and of Distribution (.#), which characterise in the ordinary 
way the conceptions of the straight line and the plane: also 
by the postulates of Congruence (C), and the Postulate of 
Archimedes (Z>). 

Relative to this fundamental set, which we shall denote 
by {A, B, C, Z>}, the following hypotheses are mutually 
equivalent, and equivalent also to that stated by EUCLID in 
his Fifth Postulate: 

a) The internal angles, which two parallels make with a 
transversal on the same side, are supplementary [Ptolemy]. 

b) Two parallel straight lines are equidistant. 

c) If a straight line intersects one of two parallels, it 
also intersects the other (Proclus); 

I2O IV. The Founders of Non-Euclidean Geometry (Contd.). 


Two straight lines, which are parallel to a third, are 
parallel to each other; 
or again, 

Through a point outside a straight line there can be 
drawn one and only one parallel to that line. 

d) A triangle being given, another triangle can be con 
structed similar to the given one and of any size whatever. 

e) Through three points, not lying on a straight line, a 
sphere can always be drawn. [W. BOLYAI.] 

f) Through a point between the lines bounding an angle 
a straight line can always be drawn which will intersect these 
two lines. [LORENZ.] 

a) If the straight line r is perpendicular to the trans 
versal AB and the straight line s cuts it at an acute angle* 
the perpendiculars from the points of s upon r are less than 
AB, on the side in which AB makes an acute angle with s. 

p) The locus of the points which are equidistant from 
a straight line is a straight line. 

f) The sum of the angles of a triangle is equal to two 
right angles. [SACCHERI.] 

Now let us suppose that we diminish the fundamental 
set of hypotheses, cutting out the Archimedean Hypothesis. 
Then the propositions (a), (b), (c), (d), (e) and (f) are 
mutually equivalent, and also equivalent to the Fifth Postu 
late of Euclid, with respect to the fundamental set {A, B, Cj . 
With regard to the propositions (a), (p), (T), while they are 
mutually equivalent with respect to the set \A, B, C} no one 
of them is equivalent to the Euclidean Postulate. This result 
brings out clearly the importance of the Postulate of Archi 
medes. It is given in the memoir of DEHN r [1900] to which 

1 Cf. Note on p. 30. 

Hypotheses Equivalent to Euclid s Postulate. 121 

reference has already been made. In that memoir it is shown 
that the hypothesis (t) on the sum of the angles of a triangle 
is compatible not only with the ordinary elementary geo 
metry, but also with a new geometry necessarily Non-Archi 
medeanwhere the Fifth Postulate does not hold, and in 
which an infinite number of lines pass through a point and 
do not intersect a given straight line. To this geometry the 
author gave the name of Semi-Euclidean Geometry. 

The Spread of Non-Euclidean Geometry. 

60. The works of LOBATSCHEWSKY and BOLYAI did 
not receive on their publication the welcome which so many 
centuries of slow and continual preparation seemed to 
promise. However this ought not to surprise us. The 
history of scientific discovery teaches that every radical change 
in its separate departments does not suddenly alter the con 
victions and the presuppositions upon which investigators 
and teachers have for a considerable time based the present 
ation of their subjects. 

In our case the acceptance of the Non-Euclidean Geo 
metry was delayed by special reasons, such as the difficulty 
of mastering LOBATSCHEWSKY S works, written as they were in 
Russian, the fact that the names of the two discoverers were 
new to the scientific world, and the Kantian conception of 
space which was then in the ascendant. 

LOBATSCHEWSKY S French and German writings helped 
to drive away the darkness in which the new theories were 
hidden in the first years; more than all availed the constant 
and indefatigable labors of certain geometers, whose names 
are now associated with the spread and triumph of Non- 
Euclidean Geometry. We would mention particularly: C. L. 
GERLING [17881864], R. BALTZER [18181887] and FR. 
SCHMIDT [1827 1901], in Germany; J. HO(JEL [1823 

122 IV. The Founders of Non-Euclidean Geometry (Contd.). 

1886], G. BATTAGLINI [1826 1894], E. BELTRAMI [1835 
1900], and A. FORTI, in France and Italy. 

61. From 1816 GERLING kept up a correspondence 
upon parallels with GAUSS T , and in 1819 he sent him 
SCHWEIKART S memorandum on Astralgeometrie [cf. p. 75]. 
Also he had heard from GAUSS himself [1832], and in terms 
which could not help exciting his natural curiosity, of a 
kleine Schrift on Non-Euclidean Geometry written by a 
young Austrian officer, son of W. BOLYAL* The bibliograph 
ical notes he received later from GAUSS [1844] on the works 
of LOBATSCHEWSKY and BOLYAI 3 induced GERLING to procure 
for himself the Geometrischen Untersuchungen and the Appen 
dix, and thus to rescue them from the oblivion into which 
they seemed plunged. 

62. The correspondence between GAUSS and SCHU 
MACHER, published between 1860 and i863, 4 tne numerous 
references to the works of LoBATSCHEWSKy and BOLYAI, and 
the attempts of LEGENDRE to introduce even into the elemen 
tary text books a rigorous treatment of the theory of pa 
rallels, led BALTZER, in the second edition of his Elemmte der 

1 Cf. GAUSS, Werke, Bd. VIII, p. 167169. 

2 Cf. GAUSS S letter to GERLING (GAUSS, Werke, Bd. VIII, 
p. 220). In this note GAUSS says with reference to the contents 
of the Appendix : "worin ich alle meine eigenen Ideen und Resultate 
iviederfinde mit grower Eleganz entwickelt^ And of the author of 
the work: ,,Ich halte diesen jungen Geometer v. Bolyai fiir ein Genie 
crster Grofie". 

3 Cf. GAUSS, Werke, Bd. VIII, p. 234-238. 

4 Briefwechsel zivischen C. F. Gauss ioid II. C. Schumacher, 
Bd. H, p. 268431 Bd. V, p. 246 (Altona, 18601863). As to 
GAUSS S opinions at this time, see also, SARTORIUS VON WALTERS" 
IIAUSKN, Gaitfi zutn GeJiichtnis, p. 80 81 (Leipzig, 1856). Cf. GAuss, 
Werke, Bd. VIII, p. 267268. 

The Spread of Non-Euclidean Geometry. 123 

Mathematik (1867), to substitute, for the Euclidean definition 
of parallels one derived from the new conception of space. 
Following LOBATSCHEWSKY he placed the equation A + B 
+ C = 1 80, which characterises the Euclidean triangle, 
among the experimental results. To justify this innovation, 
BALTZER did not fail to insert a brief reference to the possi 
bility of a more general geometry than the ordinary one, 
founded on the hypothesis of two parallels. He also gave 
suitable prominence to the names of its founders. 1 At the 
same time he called the attention of HottEL, whose interest 
in the question of elementary geometry was well known to 
scientific men, 2 to the Non-Euclidean geometry, and re 
quested him to translate the Geometrischm Untersuchungen 
and the Appendix into French. 

63. The French translation of this little book by 
LOBATSCHEWSKY appeared in 1866 and was accompanied 
by some extracts from the correspondence between GAUSS 
and SCHUMACHER. 3 That the views of LOBATSCHEWSKY, 
BOLYAI, and GAUSS were thus brought together was extremely 
fortunate, since the name of GAUSS and his approval of the 
discoveries of the two geometers, then obscure and unknown, 

1 Cf. BALTZER, Elemente der Mathematik^ Bd. II (5. Auflage) 
p. 12 14 (Leipzig, 1878). Also T. 4, p. 57, of CREMONA S trans 
lation of that work (Genoa, 1867). 

2 In 1863 HotJEL had published his wellknown Essai d nne 
exposition rationelle des principes fondamentaux de la Geomitrie ele~ 
mentaire. Archiv d. Math. u. Physik, Bd. XL (1863). 

3 Me*m. de la Soc. des Sci. de Bordeaux, T. IV, p. 88120 
(1866). This short work was also published separately under the 
title Ktttdes geomctriques siir la thcorie des paralleles par N. I. LOBAT 
SCHEWSKY, Conseiller d Etat de PEmpire de Russie et Professeur 
a 1 Universite de Kasan: traduit de 1 allemand par J. Ho DEL, sttivis 
d un Exlrait de la correspondance de Gauss et de Schumacher, (Paris, 
G. ViLLARS, 1866). 

124 ^ T ne Founders of Non-Euclidean Geometry (Contd.). 

helped to bring credit and consideration to the new doctrines 
in the most efficacious and certain manner. 

The French translation of the Appendix appeared in 
i867. x It was preceded by a Notice sur la vie et les travaux 
des deux mathematiciens hongrois W. et J. Bolyai de Hofya, 
written by the architect Fr. SCHMIDT at the invitation of 
Hoi)EL, 2 and was supplemented by some remarks by W. BOL 
YAI, taken from Vol. I of the Tentamen and from a short 
analysis, also by WOLFGANG, of the Principles of Arithmetic 
and Geometry. 3 

In the same year [1867] SCHMIDT S discoveries regard 
ing the BOLYAIS were published in the Archiv d. Math. u. 
Phys. Also in the following year A. FORTI, who had already 
written a critical and historical memoir on LoBATSCHEWsKY, 1 

1 Mem. de la Soc. des Sc. de Bordeaux, T. V, p. 189 
248. This short work was also published separately unter the 
title: La Science absolute de Fespace, independence de la verite ou 

faussete de I Axiome XI d Eudide (que ton ne pourra jamais etablir a 
priori); suivie de la quadrature geometrique du cerclc, dans le cas de 
la faussete de I Axiome XI, par Jean Bolyai, Capitaine au Corps 
du genie dans 1 armee autrichienne; Precede d une notice sur la vie 
et les travaux de W. et de J. Bolyai, par M. FR. SCHMIDT, (Paris, 
G. VILLARS, 1868). 

2 Cf. P. SxAcKEL, Franz Schmidt, Jahresber. d. Deutschen 
Math. Ver., Bd. XI, p. 141 146 (1902). 

3 This little book of W. BOLYAI S is usually referred to 
shortly by the first words of the title Kurzer Grundriss. It was pub 
lished at Maros-Vasarhely in 1851. 

4 Intorno alia geomelria immaginaria o non euclidiana. Consid- 
erazioni storico-critiche. Rivista Bolognese di scienze, lettere, arti 
e scuole, T. II, p. 171 184 (1867). It was published separately 
as a pamphlet of 16 pages (Bologna, Fava e Garagnani, 1867). 
The same article, with some additions and the title, Siudii gco- 
metrici sulla teorica delle parallele di N. J. Lobatschewky, appeared 
in the political journal La Provincia di Pisa, Anno III, Nr. 25, 27, 

Hoiiel and Schmidt. 135 

made the name and the works of the two now celebrated 
Hungarian geometers known to the Italians. 1 

To the credit of HOUEL there should also be mentioned 
his interest in the manuscripts of JOHANN BOLYAI, then [1867] 
preserved, in terms of WOLFGANG S will, in the library of 
the Reformed College of Maros-Vasarhely. By the help of 
Prince B. BONCAMPAGNI [1821 1894], who in his turn in 
terested the Hungarian Minister of Education, Baron EOTVOS, 
he succeeded in having them placed [1869] in the Hungarian 
Academy of Science at Budapest. 2 In this way they became 
more accessible and were the subject of painstaking and 
careful research, first by SCHMIDT and recently by STACKEL. 

In addition HOUEL did not fail in his efforts, on every 
available opportunity, to secure a lasting triumph for the Non- 
Euclidean Geometry. If we simply mention his Essai cri 
tique sur les principes fondamenteaux de la gtometrie:* his ar 
ticle, Sur I impossibility de demontrer par une construction 
plane le postulatwn d Euclide;* the Notices sur la vie et les 
travaux de N. f. Lob ats chew sky; s and finally his translations 
of various writings upon Non-Euclidean Geometry into French, 6 

2 9> 3 I 1 867); and part of it was reprinted under the original title 
(Pisa, Nistri, 1867). 

1 Cf. Intorno alia vita ed agli scritti di Wolfgang e Giovanni 
Bolyai di Bolya, matematid ungheresi. Boll, di Bibliografia e di 
Storia delle Scienze Mat. e Hsiche. T. I, p. 277299 (1869). 
Many historical and bibliographical notes were added to this article 
of Ford s by B. BONCOMPAGNI. 

2 Cf. STACKEL S article on Franz Schmidt referred to above. 

3 i. Ed., G. VILLARS, Paris, 1867; 2 Ed., 1883 (cf. Note 3 

P- 52)- 

4 Giornale di Mat. T. VII p. 8489; Nouvelles Annales (2) 
T. IX, p. 93-96. 

5 Bull. des. Sc. Math. T. I, p. 66-71, 324328, 384388 

6 In addition to the translations mentioned in the text, HOUEL 

126 IV- The Founders of Non-Euclidean Geometry (Contd.). 

it will ^e understood how fervent an apostle this science had 
found in the famous French mathematician. 

HOTEL S labours must have urged J. FRISCHAUF to per 
form the service for Germany which the former had rendered 
to France. His book Absolute Geometrie nach J. Bolyai 
(1872) is simply a free translation of JOHANN S Appendix, to 
which were added the opinions of W. BOLYAI on the Found 
ations of Geometry. A new and revised edition of FRISCH- 
AUF S work was brought out in 1876 2 . In that work reference 
is made to the writings of LOBATSCHEWSKY and the memoirs 
of other authors who about that time had taken up this study 
from a more advanced point of view. This volume remained 
for many years the only book in which these new doctrines 
upon space were brought together and compared. 

64. With equal conviction and earnestness GIUSEPPE 
BATTAGLINI introduced the new geometrical speculations into 
Italy and there spread them abroad. From 1867 the Gior- 
nale di Matematica, of which he was both founder and editor, 
became the recognized organ of Non-Euclidean Geometry. 

BATTAGLINI S first memoir Sulla geometria immaginaria 
di Lobatschewsky^ was written to establish directly the prin 
ciple which forms the foundation of the general theory of 
parallels and the trigonometry of LOBATSCHEWSKY. It was 

translated a paper by BATTAGLINI (cf. note 3), two by BELTRAMI 
(cf. note 2 p. 127 and p. 147); one, by RIEMANN (cf. note p. 138), 
and one by HELMHOLTZ (cf. note p. 152). 

1 (xii -f 96 pages) (Teubner, Leipzig). 

2 Elements der Absoluten Geometrie, (vi -j- 142 pages) (Teubner, 

3 Giornale di Mat. T. V, p. 217231 (1867). Rend. Ace. 
Science Fis. e Matem. Napoli, T. VI, p. 157 173 (1867). French 
translation, by HOUEL, Nouvelles Annales (2) T. VII, p. 20921, 
265277 (1868). 

Battaglini and Beltrami. 127 

followed, a few pages later, by the Italian translation of the 
Pangtometrie* ; and this, in its turn, in 1868, by the translation 
of the Appendix. 

At the same time, in the sixth volume of the Giornale di 
Matematica, appeared E. BELTRAMI S famous paper, Saggio di 
interpretazione delta geometria non euclidea. 2 This threw an 
unexpected light on the question then being debated regard 
ing the fundamental principles of geometry, and the concep 
tions of GAUSS and LoBATSCHEWSKY. 3 

Glancing through the subsequent volumes of the Giorn 
ale di Matematica we frequently come upon papers upon 
Non-Euclidean Geometry. There are two by BELTRAMI [1872] 
connected with the above named Saggio; several by BATT 
AGLINI [1874 78] and by d OviDio [1875 77L which treat 
some questions in the new geometry by the projective me 
thods discovered by CAYLEY; HOUEL S paper [1870] on the 
impossibility of demonstrating Euclid s Postulate; and others 
by CASSANI [1873 81], GUNTHER [1876], DE ZOLT [1877], 
FRATTINI [1878], RICORDI [1880], etc. 

65. The work of spreading abroad the knowledge of 
the new geometry, begun and energetically carried forward 
by the aforesaid geometers, received a powerful impulse from 
another set of publications, which appeared about this time 
[1868 72]. These regarded the problem of the foundations 
of geometry in a more general and less elementary way than 
that which had been adopted in the investigations of GAUSS, 

1 This was also published separately as a small book, entitled, 
Pangeometria o sunto di geometria fondata sopra una teoria generate 
e rigorosa delle parallele (Naples, 1867; 2a Ed. 1874). 

2 It was translated into French by HOUEL in the Ann. Sc. de 
PEcole Normale Sup., T. VI, p. 251288 (1869). 

3 Cf. Commemorazione di E. Beltrami by L. CREMONA: Giornale 
di Mat. T. XXXVIII, p. 362 (1900). Also the Nachruj by E. 
PASCAL, Math. Ann. Bd. LVII, p. 65107 (1903). 

128 IV. The Founders of Non-Euclidean Geometry (Contd.). 

LOBATSCHEWSKY, and BOLYAI. In Chapter V. we shall shortly 
describe these new methods and developments, which are asso 
ciated with the names of some of the most eminent mathe 
maticians and philosophers of the present time. Here it is 
sufficient to remark that the old question of parallels, from 
which all interest seemed to have been taken by the in 
vestigations of LEGENDRE forty years earlier, once again and 
under a completely new aspect attracted the attention of geo 
meters and philosophers, and became the centre of an 
extremely wide field of labour. Some of these efforts were 
simply directed toward rendering the works of the founders 
of Non-Euclidean geometry more accessible to the general 
mathematical public. Others were prompted by the hope of 
extending the results, the content, and the meaning of the 
new doctrines, and at the same time contributing to the pro 
gress of certain special branches of Higher Mathematics. 1 

1 Cf. e. g., E. PICARD, La Science Moderne et son etat 
actuel, p. 75 (Paris, FLAMMARION, 1905). 

Chapter V. 

The Later Development of Non-Euclidean 

66. To describe the further progress of Non-Euclidean 
Geometry in the direction of Differential Geometry and Pro- 
jective Geometry, we must leave the field of Elementary Mathe 
matics and speak of some of the branches of Higher Mathe 
matics, such as the Differential Geometry of Manifolds, the 
Theory of Continuous Transformation Groups, Pure Projec- 
tive Geometry (the system of STAUDT) and the Metrical 
Geometries which are subordinate to it. As it is not consistent 
with the plan of this work to refer, even shortly, to these 
more advanced questions, we shall confine ourselves to those 
matters without which the reader could not understand the 
motive spirit of the new researches, nor be led to that other 
geometrical system, due to RIEMANN, which has been alto 
gether excluded from the previous investigations, as they 
assume that the straight line is of infinite length. 

This system is known by the name of its discoverer and 
corresponds to the Hypothesis of the Obtuse Angle of SAC- 


1 The reader, who wishes a complete discussion of the sub 
ject of this chapter, should consult KLEIN S Vorlesungen iiber die 
)iichtenklidische Geometric, (Gottingen, 1903); and BlANCHl s Lezioni 
sulla Geometria differenziale, 2 Ed. T. I, Cap. XI XIV (Pisa, Spoerri, 
1903). German translation by LUKAT, i*t Ed. (Leipzig, 1899). Also 
The Elements of Non-Euclidean Geometry by T. L. COOLIDGE which 
has recently (1909) been published by the Oxford University Press. 


1 30 V. The Later Development of Non-Euclidean Geometry. 

Differential Geometry and Non-Euclidean Geometry. 
The Geometry upon a Surface. 

67. What follows will be more easily understood if 
we start with a few observations: 

A surface being given, let us see how far we can establish 
a geometry upon it analogous to that on the plane. 

Through two points A and B on the surface there will 
generally pass one definite line belonging to the surface, 
namely, the shortest distance on the surface between the two 
points. This line is called the geodesic joining the two points. 
In the case of the sphere, the geodesic joining two points, not 
the extremities of a diameter, is an arc of the great circle 
through the two points. 

Now if we wish to compare the geometry upon a surface 
with the geometry on a plane, it seems natural to make the 
geodesies, which measure the distances on the one surface, 
correspond to the straight lines of the other. It is also natural 
to consider two figures traced upon the surface as (geodetical- 
ly) equal, when there is a point to point correspondence be 
tween them, such that the geodesic distances between corre 
sponding points are equal. 

We obtain a representation of this conception of equality, 
if we assume that the surface is made of a flexible and i?iex-, 
tensible sheet. Then by a movement of the surface, which does 
not remain rigid, but is bent as described above, those figures 
upon it, which we have called equal, are to be superposed 
the one upon the other. 

Let us take, for example, a piece of a cylindrical surface. 
By simple bending, without stretching, folding, or tearing, this 
can be applied to a plane area. It is clear that in this case 
two figures ought to be called equal on the surface, which 
coincide with equal areas on the plane, though of course two 
such figures are not in general equal in space. 

Differential Geometry and Non-Euclidean Geometry. 13! 

Returning now to any surface whatsoever, the system of 
conventions, suggested above, leads to a geometry on the sur 
face, which we propose to consider always for suitably bounded 
regions [Normal Regions]. Two surfaces which are applicable 
the one to the other, by bending without stretching, will have 
the same geometry. Thus, for example, upon any cylindrical 
surface whatsoever, we will have a geometry similar to that on 
any plane surface, and, in general, upon any developable surface. 

The geometry on the sphere affords an example of a 
geometry on a surface essentially different from that on the 
plane, since it is impossible to apply a portion of the sphere 
to the plane. However there is an important analogy be 
tween the geometry on the plane and the geometry on the 
sphere. This analogy has its foundation in the fact that the 
sphere can be freely moved upon itself, so that propositions 
in every way analogous to the postulates of congruence on 
the plane hold for equal figures on the sphere. 

Let us try to generalize this example. In order that a 
suitably bounded surface, by bending but without stretching, 
can be moved upon itself in the same way as a plane, a cer 
tain number \K\ invariant with respect to this bending, must 
have a constant value at all points of the surface. This number 
was introduced by GAUSS and called the Curvature. 1 [In 
English books it is usually called Gauss s Curvature or the 
Measure of Curvaturel\ 

i Remembering that the curvature at any pon t of a plane 
curve is the reciprocal of the radius of the osculating circle for 
that point, we shall now show that the curvature at a point J/ of the 
surface can be defined. Having drawn the normal n to the surface 
at M, we consider the pencil of planes through ;/, and the corre 
sponding pencil of curves formed by their intersections with the 
surface. In this pencil of (plane) curves, there are two, orthogonal 
to each other, whose curvatures, as defined above, are maximum 
and minimum. The product of their curvatures is GAUSS S Curva 
ture of the Surface at M. This Curvature has one most marked 

132 V. The Later Development of Non-Euclidean Geometry. 

Surfaces of Constant Curvature can be actually con 
structed. The three cases 

have to be distinguished. 

For K = O, we have the developable surfaces [applic 
able to the plane]. 

For K^> O, we have the surfaces applicable to a sphere 
of radius i : Y A , and the sphere can be taken as a model 
for these surfaces. 

For K<^O } we have the surfaces applicable to the 
Pseudosphere, which can be taken as a model for the surfaces 
of constant negative curvature. 

Pseudosphere. Tractrix. 

Fig. 54- Fig. 55. 

The Pseudosphere is a surface of revolution. The equat 
ion of its meridian curve (the tractrix x ) referred to the axis 

characteristic. It is unchanged for every bending of the surface 
which does not involve stretching. Thus, if two surfaces are 
applicable to each other in the sense of the text, they ought to 
have the same Gaussian Curvature at corresponding points [GAUSS]. 

This result, the converse of which was proved by MINDING 
to hold for Surfaces of Constant Curvature, shows that surfaces, 
freely movable upon themselves, are characterised by constancy of 

1 The tractrix is the curve in which the distance from the 

Surfaces of Constant Curvature. 133 

of rotation 2, and to a suitably chosen axis ofx perpendicular 
to z, is 

where k is connected with the Curvature K by the equation 

To the pseudosphere generated by (i) can be applied 
any portion of the surface of constant curvature ,-. 

Surface of Constant Negative Curvature.* 

Fig. 56. 

point of contact of a tangent to the point where it cuts its 
asymptote is constant. 

1 Fig. 56 is reproduced from a photograph ef a surface con 
structed by BELTRAMI. The actual model belongs to the collection 
of models in the Mathematical Institute of the University of Pavia. 

j?4 V. The Later Development of Non-Euclidean Geometry. 

68. There is an analogy between the geometry on a 
surface of constant curvature and that of a portion of a plane, 
both taken within suitable boundaries. We can make this 
analogy clear by translating the fundamental definitions and 
properties of the one into those of the other. This is indicat 
ed shortly by the positions which the corresponding terms 
occupy in the following table: 

(a) Surface. (a) Portion of the plane. 

(b) Point. (b) Point. 

(c) Geodesic. (c) Straight line. 

(d) Arc of Geodesic. (d) Rectilinear Segment. 

(e) Linear properties of the (e) Postulates of Order for 
Geodesic. points on a Straight Line. 

(f) A Geodesic is determined (f) A Straight Line is deter- 
by two points. mined by two points. 

(g) Fundamental properties (g) Postulates of Congruence 
of the equality of Geode- for Rectilinear Segments 
sic Arcs and Angles. and Angles. 

(h) If two Geodesic triangles (h) If two Rectilinear triang- 
have their two sides and les have their two sides 

the contained angles e- and the contained angles 

qual, then the remaining equal, then the remaining 

sides and angles are equal. sides and angles are equal. 

It follows that we can retain as common to the geome 
try of the said surfaces all those properties concerning bound 
ed regions on a plane, which in the Euclidean system are 
independent of the Parallel Postulate, when no use is made 
of the complete plane [e. g., of the infinity of the straight 
line] in their demonstration. 

We must now proceed to compare the propositions for 
a bounded region of the plane, depending on the Euclidean 
hypothesis, with those which correspond to them in the geo 
metry on the surface of constant curvature. We have, e. g., 
the proposition that the sum of the angles of a triangle is 

Geometry on a Surface of Constant Curvature. jsc 

equal to two right angles. The corresponding property does 
not generally hold for the surface. 

Indeed GAUSS showed that upon a surface whose curva 
ture K is constant or varies from point to point, the surface 

over the whole surface of a geodesic triangle ABC, is equal 
to the excess of its three angles over two right angles. I 

i. e. || KdS = A + B + C TT. 


Let us apply this formula to the surfaces of constant 
curvature, distinguishing the three possible cases 
Case I. K=O. 

In this case we have 

KdS = O- that is A + B + C= TT. 


Thus the sum of the angles of a geodesic triangle on sur 
faces of zero curvature is equal to two right angles. 

Case II. K= l k ^> O. 

In this case we have 


But \\dS=- area of the triangle ABC= A. 

From this equation it follows that 
A + B + C> TT, 
and that A = k 2 (A + B + C TT). 

i Cf. BIANCHI S work referred to above; Chapter VI. 

136 V. The Later Development of Non-Euclidean Geometry. 

That is: 

a) The sum of the angles of a geodesic triangle on sur 
faces of constant positive curvature is greater than two right 

b) The area of a geodesic triangle is proportional to the 
excess of the sum of its angles over two right angles. 

Case III. K= -*- < O 

In this case we have 


where we again denote the area of the triangle ABC by A. 
Then we have 

= TT (A + B + C). 

From this it follows that 

A + + C<TT, 

and that A = k 2 (TT A- BC}. 

That is: 

a) The sum of the angles of a geodesic triangle on sur 
faces of constant negative curvature is less than two right angles. 

b) The area of a geodesic triangle is proportional to the 
difference between the sum of its angles and two right angles. 

We bring these results together in the following table: 

Surfaces of Constant Curvature. 

Value of the Curvature 

of the Surface 

Character of the Triangle 




The Geodesic Triangle. 137 

With the geometry of surfaces of zero curvature and of 
surfaces of constant positive curvature we are already ac 
quainted, since they correspond to Euclidean plane geometry 
and to spherical geometry. 

The study of the surfaces of constant negative curvature 
was begun by F. MINDING [1806 1885] with the investiga 
tion of the surfaces of revolution to which they could be ap 
plied. 1 The following remark of MINDING S, fully proved 
by D. CODAZZI [1824 1873], establishes the trigonometry 
- of such surfaces. In the formulae of spherical trigonometry let 
the angles be kept fixed and the sides multiplied by i = J/ i. 
Then we obtain the equations which are satisfied by the elements 
of the geodesic triangles on the surfaces of constant negative cur 
vature* These equations [the pseudospherical trigonometry} 
evidently coincide with those found by TAURINUS; in other 
words, with the formulae of the geometry of LOBATSCHEWSKY- 

69. From the preceding paragraphs it will be seen that 
the theorems regarding the sum of the angles of a triangle in 
the geometry on surfaces of constant curvature, are related to 
those of plane geometry as follows: 

For K= O they correspond to those which hold on the 
plane in the case of the Hypothesis of the Right Angle. 

For K^> O they correspond to those which hold on the 
plane in the case of the Hypothesis of the Obtuse Angle. 

1 Wie sich entscheiden ldsst t ob zzvei gegebene kntmme Fldchen 
aufeinander abwickelbar sind oder nicht\ ncbst Bemerkungen tiber die 
Flachcn von unveranderlichem A rumrmengsmasse. CRELLE S Journal, 
Bd. XIX, p. 370-387 (I8 3 9). 

2 MINDING: Beitriige zur Theorie der kiirzesteii Linien aitf krummen 
Fldchen. CRELLE S Journal, Bd. XX, p. 323327 (1^40). D. CODAZZI: 
Intonto alle superficie^ le quali hanno costante il prodotto d<; due raggi 
di cnnjaticra. Ann. di Scienze Mat. e Fis. T. VIII, p. 346355 

1^8 V- The Later Development of Non-Euclidean Geometry. 

For K<^O they correspond to those which hold on the 
plane in the case of the Hypothesis of the Acute Angle. 

The first of the results is evident a priori, since we are 
concerned with developable surfaces. 

The analogy between the geometry of the surfaces of con 
stant negative curvature, for example, and the geometry of 
LoBATSCHEWSKY-BoLYAi, could be made still more evident by 
arranging in tabular form the relations between the elements 
of the geodesic triangles traced upon those surfaces, and the 
formulae of Non-Euclidean Trigonometry. Such a comparison 
was made by E. BELTRAMI in his Saggio di interpretazione delta 
geometria non-euclidea. T 

In this way it will be seen that the geometry upon a sur 
face of constant positive or negative curvature can be con 
sidered as a concrete interpretation of the Non-Euclidean Geo 
metry, obtained in a bounded plane area, with the aid of the 
Hypothesis of the Obtuse Angle or that of the Acute Angle. 

The possibility of interpreting the geometry of a two- 
dimensional manifold by means of ordinary surfaces was ob 
served by B. RIEMANN [1826 1866] in 1854, the year in 
which he wrote his celebrated memoir: t)ber die Hypothesen 
welche der Geometrie zugrunde liegen. 2 The developments of 

1 Giorn. di Mat, T. VI, p. 284312 (1868). Opere Mat., 
T. I, p. 374405 (Hoepli, Milan, 1902). 

2 Riemanns Werke , I. Aufl. (1876), p. 254312: 2. Aufl. 
(1892), p. 272287. It was read by RIEMANN to the Philosophical 
Faculty at Gottingen as his Habilitationsschrift, before an audience 
not composed solely of mathematicans. For this reason it does 
not contain analytical developments, and the conceptions intro 
duced are mostly of an intuitive character. Some analytical ex 
planations are to be found in the notes on the Memoir sent by RIE 
MANN as a solution of a problem proposed by the Paris Academy 
(Riemanns Werke, I. Aufl., p. 384391). The philosophical basis 
of the Habilitationsschrift is the study of the properties of things 
from their behaviour as infinitesimals. Cf. KLEIN S discourse; 

Beltrami and Riemann. I -jg 

Non-Euclidean Geometry in the direction of Differential Ge 
ometry are directly due to this memoir. 

BELTRAMI S interpretation appears as a particular case of 
RIEMANN S. It shows clearly, from the properties of surfaces 
of constant curvature, that the chain of deductions from the 
three hypotheses regarding the sum of the angles of a triangle 
must lead to logically consistent systems of geometry. 

This conclusion, so far as regards the Hypothesis of the 
Obtuse Angle, seems to contradict the theorems of SACCHERI, 
LAMBERT, and LEGENDRE, which altogether exclude the possi 
bility of a geometry founded on that hypothesis. However 
the contradiction is only apparent. It disappears if we remem 
ber that in the demonstration of these theorems, not only 
the fundamental properties of the bounded plane are used, but 
also those of the complete plane, e. g., the property that the 
straight line is infinite. 

Principles of Plane Geometry on the Ideas of 

70. The preceding observations lead us to the foun 
dation of a metrical geometry, which excludes Euclid s Postul- 

Riemanu und seine Bedeiitung in der Entwickelung der modernen 
Mathematik. Jahresb. d. Deutschen Math. Ver., Bd. IV, p. 7282 
(1894), and the Italian translation by E.PASCAL in Ann. di Mat., (2), 
T. XXIII, p. 222. The Habilitationsschrift was first published in 1867 
after the death of the author [Gott. Abh. XIII] under the editor 
ship of DEDEKIND. It was then translated into French by J. HOUEL 
[Ann. di Mat. (2). T. Ill (1870), Oeuvres de Riemann, (1876)]; into 
English, by W. K. CLIFFORD [Nature, Vol. VIII, (1873)], and again 
by G. B. HALSTED [Tokyo sagaku butsurigaku kwai kiji, Vol. VII, 
(1895); into Polish, by DICKSTEIN (Comm. Acad. Litt. Cracov. 
Vol. IX, 1877); into Russian, by D. SINTSOFF [Mem. of the Phy 
sical Mathematical Society of the University of Kasan, (2), Vol. Ill, 

140 V Tlie Later Development of Non-Euclidean Geometry. 

ate, and adopts a more general point of view than that for 
merly held: 

(a) We assume that we start from a bounded plane area 
(normal region], and not from the whole plane. 

(b) We regard as postulates those elementary propositions, 
which are revealed to us by the setises for the region originally 
taken; the propositions relative to the straight line being determ 
ined by two points, to congruence, etc. 

(c) We assifme that the properties of the initial region can 
be extended to the neighbourhood of any point on the plane \ivc 
do not say to the complete plane, viewed as a whole]. 

The geometry, built upon these foundations, will be the 
most general plane geometry, consistent with the data which 
rigorously express the result of our experience. These results 
are, however, limited to an accessible region. 

From the remarks in 69, it is clear that the said geo 
metry will find a concrete interpretation in that of the sur 
faces of constant curvature. 

This correspondence, however, exists only from the 
point of view (differential} according to which only bounded 
regions are compared. If, on the other hand, we place our 
selves at the (integral} point of view, according to which the 
geometry of the whole plane and the geometry on the sur 
face are compared, the correspondence no longer exists. In 
deed, from this standpoint, we cannot even say that the same 
geometry will hold on two surfaces with the same constant 
curvature. For example, a circular cylinder has a constant 
curvature, zero, and a portion of it can be applied to a region 
of a plane, but the entire cylinder cannot be applied in this 
way to the entire plane. The geometry of the complete cy 
linder thus differs from that of the complete Euclidean plane. 
Upon the cylinder there are closed geodesies (its circular 
sections), and, in general, two of its geodesies (helices) meet 
in an infinite number of points, instead of in just two. 

Riemann s New Geometry. \A \ 

Similar differences will in general appear between a me 
trical Non-Euclidean geometry, founded on the postulates 
enunciated above, and the geometry on a corresponding sur 
face of constant curvature. 

When we attempt to consider the geometry on a surface 
of constant curvature (e. g., on the sphere or pseudosphere) 
as a whole, we see, in general, that the fundamental property 
of a normal region that a geodesic is fully determined by two 
points ceases to hold. This fact, however, is not a necessary 
consequence of the hypotheses on which, in the sense above 
explained, a general metrical Non-Euclidean geometry of the 
plane is based. Indeed, when we examine whether a system 
of plane geometry is logically possible, which will satisfy the 
conditions (a), (b), and (c), and in which the postulates of con 
gruence and that a straight line is fully determined by two 
points are valid on the complete plane, we obtain, in addition 
to the ordinary Euclidean system, the two following systems 
of geometry: 

1. The system of Lobatschewsky-Bolyai, already explain 
ed, in which two parallels to a straight line pass through a 

2. A neiv system (called Riemann s system} which cor 
responds to SACCHERI S Hypothesis of the Obtuse Ang/e, and 
in which no parallel lines exist. 

In the latter system the straight line is a closed line of 
finite length. We thus avoid the contradiction to which we 
would be led if we assumed that the straight line were open 
(infinite). This hypothesis is required in proving Euclid s The 
orem of the Exterior Angle [I. 16] and some of SACCHERI S 

71. RIEMANN was the first to recognize the existence 
cf a system of geometry compatible with the Hypothesis of 
the Obtuse Angle, since he was the first to substitute for the 

142 V. The Later Development of Non-Euclidean Geometry. 

hypothesis that the straight line is infinite, the more general 
one that it is unbounded. The difference, which presents it 
self here, between infinite and unbounded is most important. 
We quote in regard to this RIEMANN S own words: 1 

In the extension of space construction to the infinitely 
great, we must distinguish between unboundedness and infinite 
extent; the former belongs to the extent relations; the latter to 
the measure relations. That space is an unbounded three-fold 
manifoldness is an assumption which is developed by every 
conception of the outer world; according to which every in 
stant the region of real perception is completed and the pos 
sible positions of a sought object are constructed, and which 
by these applications is for ever confirming itself. The un 
boundedness of space possesses in this way a greater empiri 
cal certainty than any external experience, but its infinite ex 
tent by no means follows from this; on the other hand, if we 
assume independence of bodies from position, and therefore 
ascribe to space constant curvature, it must necessarily be 
finite, provided this curvature has ever so small a positive 

Finally, the postulate which gives the straight line an in 
finite length, implicitly contained in the work of preceding 
geometers, is to RIEMANN as fit a subject of discussion as that 
of parallels. What RIEMANN holds as beyond discussion is 
the unbotmdedness of space. This property is compatible with 
the hypothesis that the straight line is infinite (open), as well 
as with the hypothesis that it is finite (closed). 

The logical possibility of RIEMANN S system can be de 
duced from its concrete interpretation in the geometry of the 
sheaf of lines. The properties of the sheaf of lines are trans- 

1 [This quotation is taken from CLIFFORD S translation in 
Nature, referred to above. (Teil III, 8 2 of RIEMANN S Memoir.)]. 

The Geometry of the Sheaf. 


lated readily into those of RIEMANN S plane, and vice versa, 
with the aid of the following dictionary : 



Plane [Pencil] 

Angle between two Lines 

Dihedral Angle 




Straight line 




We now give, as an exam 
of the best known propositions 

a) The sum of the three 
dihedral angles of a trihedron 
is greater than two right 
dihedral angles. 

b) All the planes which are 
perpendicular to another 
plane pass through a straight 

c) With every plane of 
the sheaf let us associate the 
straight line in which the 
planes perpendicular to the 
given plane all intersect. In 
this way we obtain a corres 
pondence between planes and 
straight lines which enjoys 
the following property: The 
straight lines corresponding 
to the planes of a pencil 
[Ebenenbtischel, set of planes 
through one line, the axis of 
the pencil] lie on a plane, 

pie, the translation of some 
for the sheaf: 

a) The sum of the three 
angles of a triangle is greater 
than two right angles. 

b) All the straight lines 
perpendicular to another 
straight line pass through a 

c) With every straight line 
in the plane let us associate 
the point in which the lines 
perpendicular to the given 
line intersect. In this way we 
obtain a correspondence be 
tween lines and points, which 
enjoys the following pro 

The points corresponding 
to the lines of a pencil lie on 
a straight line, which in its turn 
has for corresponding point 
the vertex of the pencil. 

144 V> The Later Development of Non-Euclidean Geometry. 

which in its turn has for cor- The correspondence thus 
responding line the axis of denned is called absolute po- 
the pencil. The correspond- larity of the plane, 
ence thus denned is called 
absolute [orthogonal] polarity 
of the sheaf. 

72. A remarkable discovery with regard to the Hypo 
thesis of the Obtuse Angle was made recently by DEHN. 

If we refer to the arguments of SACCHERI [p. 37], 
LAMBERT [p. 45], LEGENDRE [p. 56], we see at once that 
these authors, in their proof of the falsehood of the Hypo 
thesis of the Obtuse Angle, avail themselves, not only of the 
hypothesis that the straight line is infinite, but also of the 
Archimedean Hypothesis. Now we might ask ourselves if this 
second hypothesis is required in the proof of this result. In 
other words, we might ask ourselves if the two hypotheses, 
one of which attributes to the straight line the character of 
open lines, while the other attributes to the sum of the angles 
of a triangle a value greater than two right angles, are com 
patible with each other, when the Postulate of Archimedes is 
excluded. DEHN gave an answer to this question in his 
memoir quoted above (p. 30), by the construction of a Non- 
Archimedean geometry, in which the straight line is open, 
and the sum of the angles of a triangle is greater than two 
right angles. Thus the second of SACCHERI s three hypotheses 
is compatible with the hypothesis of the open straight line 
in the sense of a Non- Archimedean system. This new 
geometry was called by DEHN Non-Legendrean Geometry [cf. 
S 59, P- 121]. 

73. We have seen above that the geometry of a 
surface of constant curvature (positive or negative) does not 
represent, in general, the whole of the Non-Euclidean geo- 

Hilbert s Theorem. 145 

metry on the plane of LOBATSCHEWKY and of RIEMANN. The 
question remains whether such a correspondence could not 
be effected with the help of some particular surface of this 

The answer to this question is as follows : 
i) There does not exist any regular* analytic surface 
on which the geometry of Lobatschewsky-Bolyai is altogether 
valid [HILBERT S Theorem]. 2 

1 In other words, free from singularities. 

2 Uber Fldchen von konstanter Gaussscher Kriimmung. Trans. 
Amer. Math. Soc. Vol. II, p. 8699 (1901); Grundlagen der Geo 
metric, 2. Aufl. p. 162175. (Leipzig, Teubner, 1903). 

This question, which HILBERT S Theorem answers, was first 
suggested to mathematicians by BELTRAMI S interpretation of the 
lecture, Uber Ur sprung und Bedeutung der geometrischen Axiom c, 
(Vortrage und Reden, Bd. II. Brunswick, 1844) had denied the 
possibility of constructing a pseudospherical surface, extending 
indefinitely in every direction. Also A. GENNOCCHI in his Lettre 
a M. Quetelet sur diverges questions mathematiques t [Belgique Bull. (2). 
T. XXXVI, p. 181198 (1873)], and more fully in his Memoir, 
Sur une memoir e de D. Foncenex et sur Us geometries non-euclidiennes, 
[Torino Memorie (3), T. XXIX, p. 365 404 (1877)], showed the 
insufficiency of some intuitive demonstrations, intended to prove 
the concrete existence of a surface suitable for the representation 
of the entire Non- Euclidean plane. Also he insisted upon the 
probable existence of singular points (as for example, those on 
the line of regression of Fig. 54) in every concrete model of a 
surface of constant negative curvature. 

So far as regards HILBERT S Theorem, we add that the 
analytic character of the surface, assumed by the author, has been 
shown to be unnecessary. Cf. the dissertation of G. LUTKEMEYER: 
Uber den analytischen Charakter der Integrate -von partiellen Differ en- 
tialgleichungen, (Gotlingen, 1902). Also the Note by E. HOLMGREN: 
Sur les surfaces a courbiire constant e negative, [Comptes Rendus, I Sem., 
p. 840843 (1902)]. 

[In a recent paper Sur les surfaces a courbure constante negative, 
(Bull. Soc. Math, de France, t. XXXVII p. 5158, 1909) E. GOURSAT 


146 V. The Later Development of Non-Euclidean Geometry. 

2) A surface on which the geometry of the plane of 
Riemann would be altogether valid must be a closed surface. 

The only regular analytic closed surface of constant posi 
tive curvature is the sphere [LIEBMANN S Theorem]. 1 

But on the sphere, in normal regions of which RIEMANN S 
geometry is valid, two lines always meet in two (opposite) 

We therefore conclude that: 

In ordinary space there are no surfaces which satisfy in 
their complete extent all the properties of the Non-Euclidean 

74. At this place it is right to observe that the sphere, 
among all the surfaces whose curvature is constant and different 
from zero, has a characteristic that brings it nearer to the 
plane than all the others. Indeed the sphere can be moved 
upon itself just as the plane, so that the properties of con 
gruence are valid not only for normal regions, but, as in the 
plane, for the surface of the sphere taken as a whole. 

This fact suggests to us a method of enunciating the 
postulates of geometry, which does not exclude, a priori, the 
possible existence of a plane with all the characteristics of 
the sphere, including that of opposite points. We would 

has discussed a problem slightly less general than that enunciated 
by HILBERT, and has succeeded in proving in a fairly simple 
manner the impossibility of constructing an analytical surface of 
constant curvature, which has no singular points at a finite distance.] 
i Eine neue Eigenschajt der Kngel, Gott. Nachr. p. 44 54 
(1899). This property is also proved by HJLBERT on p. 172175 
of his Gmndlagen der Geometrie. We notice that the surfaces of 
constant positive curvature are necessarily analytic. Cf. LUTKE- 
MEYER S Dissertation referred to above (p. 163), and the memoir 
by HOLMGREN: Uber eine Klasse von partiellen Differcntialgleichungen 
der zweiten Ordnung, Math. Ann. Bd. LVII, p. 407 420 (1903). 

The Elliptic and Spherical Planes. \^n 

need to assume that the following relations were true for 
the plane: 

1) The postulates (b), (c) [cf. 70] in every normal 

2) The postulates of congruence in the whole of the 

Thus we would have the geometrical systems of EUCLID, 
of LOBATSCHEWSKY-BOLYAI, and of RIEMANN (the elliptic type), 
which we have met above, where two straight lines have 
only one common point: and a second RIEMANN S system 
(the spherical type), where two straight lines have always two 
common points. 

75- We cannot be quite certain what idea RIEMANN 
had formed of his complete plane, whether he had thought 
of it as the elliptic plane, or the spherical plane, or had 
recognized the possibility of both. This uncertainty is due 
to the fact that in his memoir he deals with Differential 
Geometry and devotes only a few words to the complete 
forms. Further, those who continued his labours in this direc 
tion, among them BELTRAMI, always considered RIEMANN S 
geometry in connection with the sphere. They were thus led 
to hold that on the complete RIEMANN S plane, as on the 
sphere (owing to the existence of the opposite ends of a 
diameter), the postulate that a straight line is determined by 
two points had exceptions, 1 and that the only form of the 
plane compatible with the Hypothesis of the Obtuse Angle 
would be the spherical plane. 

1 Cf. for example, the short reference to the geometry of 
space of constant positive curvature with which BELTRAMJ. concludes 
his memoir: Teoria fondamentale degli spazii di curvatura costantt , 
Ann. di Mat. (2). T. II, p. 354 355 (1868); or the French trans 
lation of this memoir by J. HOUEL, Ann. Sc. d. l cole Norm. Sup. 
T. VI, p. 347-377- 


148 V. The Later Development of Non-Euclidean Geometry. 

The fundamental characteristics of the elliptic plane 
were given by A. CAYLEY [1821 1895] in 1859, but the 
connection between these properties and Non-Euclidean 
geometry was first pointed out by KLEIN in 1871. To KLEIN 
is also due the clear distinction between the two geometries 
of RIEMANN, and the representation of the elliptic geometry 
by the geometry of the sheaf [cf. 871]- 

To make the difference between the spherical and 
elliptic geometries clearer, let us fix our attention on two 
classes of surfaces presented to us in ordinary space: the 
surface with two faces (two-sided] and the surface with one 
face (one-sided). 

Examples of two-sided surfaces are afforded by the 
ordinary plane, the surfaces of the second order (conicoidal, 
cylindrical, and spherical), and in general all the surfaces 
enclosing solids. On these it is possible to distinguish two 

An example of a one-sided surface is given by the 
Leaf of MOBIUS [MoBiussche Blatt], which can be easily 
constructed as follows: Cut a rectangular strip ABCD. In 
stead of joining the opposite sides AB and CD and thus 
obtaining a cylindrical surface, let these sides be joined 
after one of them, e. g., CD, has been rotated through two 
right angles about its middle point. Then what was the 
upper face of the rectangle, in the neighbourhood of CD, 
is now succeeded by the lower face of the original rectangle. 
Thus on Miibius Leaf the distinction between the two 
faces becomes impossible. 

If we wish to distinguish the one-sided surface from the 
wo-sided by a characteristic, depending only on the intrinsic 
properties of the surface, we may proceed thus: We fix a 
point on the surface, and a direction of rotation about it. 
Then we let the point describe a closed path upon the sur 
face, which does not leave the surface; for a two-sided sur- 

A One-Sided Surface. 


face the point returns to its initial position and the final 
direction of rotation coincides with the initial one; for a one 
sided surface, [as can be easily verified on the Leaf of MOBIUS, 
when the path coincides with the diametral line] there exist 
closed paths for which the final direction of rotation is oppos 
ite to the initial direction. 

Coming back to the two RIEMANN S 
planes, we can now easily state in what 
their essential difference consists: the spher 
ical plane has the character of the two-sided 
surface, and the elliptic plane that of the one 
sided surface. 

The property of the elliptic plane here The Leaf of M5bius - 
enunciated, as well as all its other propert 
ies, finds a concrete interpretation in the sheaf of lines. In 
fact, if one of the lines of the sheaf is turned about the vertex 
through half a revolution, the two rotations which have this 
line for axis are interchanged. 

Another property of the elliptic plane, allied to the 
preceding, is this: The elliptic plane, unlike the Euclidean 
plane and the other Non-Euclidean planes, is not divided by 
its lines into two parts. We can state this property other 
wise : If two points A and A are given upon the plane, and 
an arbitrary straight line, we can pass from A to A by a 
path which does not leave the plane and does not cut the 
line. 1 This fact is translated by an obvious property of the 
sheaf, which it would be superfluous to mention. 

76. The interpretation of the spherical plane by the 
sheaf of rays (straight lines starting from the vertex) is ana 
logous to that given above for the elliptic plane. The trans- 

1 A surface which completely possesses the properties of the 
elliptic plane was constructed by W. BOY. [Gott. Berichte, p. 20 
23 (1900); Math. Ann. Bd. LVII, p. 151 184 (1903)]. 

ICQ V. The Later Development of Non-Euclidean Geometry. 

lation of the properties of this plane into the properties of 
the sheaf of rays is effected by the use of a dictionary 
similar to that of 7 1 , in which the word point is found 
opposite the word ray. 

The comparison of the sheaf of rays with the sheaf of 
lines affords a useful means of making clear the connections, 
and revealing the differences, which are to be found in the 
two geometries of RIEMANN. 

We can consider two sheaves, with the same vertex, the 
one of lines, the other of rays. It is clear that to every line 
of the first correspond two rays of the second; that every 
figure of the first is formed by two symmetrical figures of the 
second; and that, with certain restrictions, the metrical pro 
perties of the two forms are the same. Thus if we agree to 
regard the two opposite rays of the sheaf of rays as forming 
one element only, the sheaf of rays and the sheaf of lines 
are identical. 

The same considerations apply to the two RIEMANN S 
planes. To every point of the elliptic plane correspond 
two distinct and opposite points of the spherical plane; to 
two lines of the first, which pass through that point, corres 
pond two lines of the second, which have two points in 
common; etc. 

The elliptic plane, when compared with the spherical 
plane, ought to be regarded as a double plane. 

With regard to the elliptic plane and the spherical 
plane, it is right to remark that the formulae of absolute tri 
gonometry, given in 56, can be applied to them in every 
suitably bounded region. This follows from the fact, al 
ready noted in 58, that the formulae of absolute trigonom 
etry hold on the sphere, and the geometry of the sphere, so 
far as regards normal regions, coincides with that of these 
two planes. 

Riemann s Solid Geometry. I tj I 

Principles of Riemann s Solid Geometry. 

77. Returning now to solid geometry, we start from 
the philosophical foundation that the postulates, although 
we grant them, by hypothesis, an actual meaning, express 
truths of experience, which can be verified only in a bounded 
region. We also assume, that on the foundation of these postul 
ates points in space are represented by three coordinates. 

On such an (analytical) representation, every line is 
given by three equations in a single variable: 

*x =/i Wi ** =/2 (0, *3 =/3 W, 

and we must now proceed to determine a function j, of 
the parameter t, which shall express the length of an arc of 
the curve. 

On the strength of the distributive property, by which 
the length of an arc is equal to the sum of the lengths of 
the parts into which we imagine it to be divided, such a 
function will be fully determined when we know the element 
of distance (ds) between two infinitely near points, whose 
coordinates are 

x l + dx,. , x 2 + dx 2 , x, 

RIEMANN starts with very general hypotheses, which 
are satisfied most simply by assuming that ds 2 , the square 
of the element of distance, is a quadratic expression in 
volving the differentials of the variables, which always re 
mains positive: 

ds 2 = Zdr/y dxi dxj, 
where the coefficients a- tj are functions of x lt x 2 , x y 

Then, admitting the principle of superposition of figures, 
it can be shown that the function a,y must be such that, with 
the choice of a suitable system of coordinates, 


152 V. The Later Development of Non-Euclidean Geometry. 

In this formula the constant K is what RIEMANN, by an ex 
tension of GAUSS S conception, calls the Curvature of Space. 

According as K is greater than, equal to, or less than 
zero, we have space of constant positive curvature, space 
of zero curvature, or space of constant negative curvature. 

We make another forward step when we assume that the 
principle of superposition [the principle of movement] can be 
extended to the whole of space, as also the postulate that a 
straight line is always determined by two points. In this way 
we obtain three forms of space; that is, three geometries 
which are logically possible, consistent with the data from 
which we set out. 

The first of these geometries, corresponding to positive 
curvature, is characterised by the fact that RIEMANN S system 
is valid in every plane. For this reason space of positive 
curvature will be unbounded and finite in all directions. 
The second, corresponding to zero curvature, is the ordinary 
Euclidean geometry. And the third, which corresponds to 
negative curvature, gives rise in every plane to the geometry 


The Work of Helmholtz and the Investigations 
of Lie. 

78. In some of his philosophical and mathematical 
writings, 1 HELMHOLTZ [1821 1894] has also dealt with the 

i Uber die thatsachlichen Grundlagen der Geometric, Heidelberg, 
Verb. d. naturw.-med. Vereins, Ed. IV, p. 197 202 (1868); Bd. V, 
p. 3132 (1869). Wiss. Abhandlungen von H. HELMHOLTZ, Bd. II, 
p. 610617 (Leipzig, 1883). French translation by J. HOUEL in 
Me"m. de la Soc. des Sc. Phys. et Nat. de Bordeaux, T. V, (18681, 
and also, in book form, along with the Etudes Gtometriques of 
LOBATSCHEWSKY and the Correspondence de Gauss et de Schumacher, 
Paris, Hermann, 1895). 

Uber die Thatsachen, die der Geometrie zum Grunde liegen. Gott. 

Helmholtz and Lie. 


question of the foundations of geometry. Instead of assum 
ing a priori the form 

ds* = lay dxi dxj, 

as the expression for the element of distance, he showed 
that this expression, in the form given to it by RIEMANN for 
space of constant curvature, is the only one possible, when, 
in addition to RIEMANN S hypotheses, we accept, from the 
beginning, that of the mobility of figures, as it would be given 
by the movement of Rigid Bodies. 

The problem of RIEMANN-HELMHOLTZ was carefully 
examined by S. LIE [1842 1899]. He started from the 
fundamental idea, recognized by KLEIN in HELMHOLTZ S 
work, that the congruence of two figures signifies that they are 
able to be transformed the one into the other, by means of a 
certain point transformation in space: and that the properties, 
in virtue of which congruence takes the logical character of 
equality , depend upon the fact that displacements are given by 
a group of transformations. 1 

In this way the problem of RIEMANN-HELMHOLTZ was 
reduced by LIE to the following form : 

Nachr. Bd. XV, p. 193221 (1868). Wiss. Abhandl., Bd. II, p. 618 


The Axioms of Geometry. The Academy, Vol. I, p. 123181 
(1870); Revue des cours sclent., T. VII, p. 498501 (1870). 

Uber die Axiome der Geometric. Populare wissenschaftliche Vor- 
trage, Heft 3, p. 2154. (Brunswick, 1876). English translation; 
Mind, Vol. I, p. 301 321. French translation; Revue scientifique 
de la France et de 1 Etranger (2). T. XII, p. 11971207 (1877) 

Uber den Ur sprung t Sinn, und Bedeutung der geometrischen 
Satze, Wiss. Abh. Bd. II, p. 640660. English translation; Mind, 
Vol. II, p. 212224(1878). 

1 Cf. Klein : Vergleichende Betrachtnngen uber neuere geometrische 
Forschutigen, (Erlangen, 1872); reprinted in Math. Ann. Bd. XLIII, 
p. 63 loo (1893). Italian translation by G. FANO, Ann. di Mat. (2), 
T. XVII, p. 301-343 (1899). 

I ZA V. The Later Development of Non-Euclidean Geometry. 

/To determine all the continuous groups in space which, 
in a bounded region, have the property of displacements. 

When these properties, which depend upon the free 
mobility of line and surface elements through a point, are 
put in a suitable form, there arise three types of groups, 
which characterise the three geometries of EUCLID, of 


Projective Geometry and Non-Euclidean Geometry. 

Subordination of Metrical Geometry to Projective 


79. In conclusion, there is an interesting connection 
between Projective Geometry and the three geometrical 

To give an idea of this last method of treating the 
question, we must remember that Projective Geometry, in 
the system of G. C. STAUDT [1798 1867], rests simply upon 
graphical notions on the relations between points, lines 
and planes. Every conception of congruence and movement 
[and thus of measurement etc.,] is systematically banished. 
For this reason Projective Geometry, excluding a certain 
group of postulates, will contain a more restricted number of 
general properties, which for plane figures are the [projecting] 
properties, remaining invariant by projection and section. 

However, when we have laid the foundations of Pro 
jective Geometry in space, we can introduce into this system 

1 Cf. LIE : Theorie der Transformalionsgnippen. Bd. Ill, p. 437 
543 (Leipzig, 1893). In connection with the same subject, H. 
PoiNCARK, in his memoir: Sur les hypotheses jondamentaux de la 
gtomltrie [Bull, de La Soc. Math, de France. T. XV, p. 203216 
U877)], solved the problem of finding all the hypotheses, which 
distinguish the fundamental group of plane Euclidean Geometry 
from the other transformation groups. 

Projective Geometry and Non-Euclidean Geometry. jcr 

the metrical conceptions, as relations between its figures and 
certain definite (metrical) entities. 

Keeping to the case of the Euclidean plane, let us see 
what graphical interpretation can be given to the fundamental 
metrical conceptions of parallelism and of perpendicularity. 

To this end we must specially consider the line at infin 
ity of the plane, and the absolute involution which the set of 
orthogonal lines of a pencil determine upon it. The double 
points of such an involution, conjugate imaginaries, are 
called the circular points (at infinity), since they are common 
to all circles in the plane [PONCELET, i822 x ]. 

On this understanding, the parallelism of two lines is 
expressed graphically by the property which they possess of 
meeting in a point on the line at infinity : the perpendicularity 
of two lines is expressed graphically by the property that 
their points at infinity are conjugate in the absolute involution, 
that is, form a harmonic range with the circular points. 
[CHASLES, i85o. 2 ] 

Other metrical properties, which can be expressed 
graphically, are those relative to the size of angles, since 
every equation 

F(A,B, ...} = O, 
between the angles A, B^ C, . . ., can be replaced by 

log a log b log c 


in which a, b, c . . . are the anharmonic ratios of the pencils 
formed by the lines bounding the angles and the (imaginary) 
ines joining the angular points to the circular points. [LA- 
GUERRE, 1853.3] 

1 Traite des proprietes projeclives dcsfigiocs. 2. Ed., T.I. Nr. 94, 
p. 48 (Paris, G. Villars, 1865). 

2 Traite de Geometric superieure. 2. Ed., Nr. 660, p. 425 (Paris, 
G. Villars, 1880). 

3 Sur la theorie des foyers. Nouv. Ann. T. XII, p. 57- Oeuvres 
de I.aguerre. T. II, p. 1213 (Paris, G. Villars, 1902). 

I eg V. The Later Development of Non-Euclidean Geometry. 

More generally it can be shown that the congruence 
of any two plane figures can be expressed by a graphical 
relation between them, the line at infinity, and the absolute 
involution. 1 Also, since congruence is the foundation of all 
metrical properties, it follows that the line at infinity and the 
absolute involution allow all the properties of Euclidean 
metrical geometry to be subordinated to Projective Geo 
metry. Thus the metrical properties appear in protective geometry, 
not as graphical properties of the figures considered in them- 
selves, but as graphical properties with regard to the funda 
mental metrical entities, made up of the line at infinity and the 
absolute involution. 

The complete set of fundamental metrical entities is 
called the absolute of the plane (CAYLEY). 

All that has been said with regard to the plane can 
naturally be extended to space. The fundamental metrical 
entities in space, which allow the metrical properties to be 
subordinated to the graphical, are the plane at infinity and a 
certain polarity (absolute polarity] on this plane. This polar 
ity is given by the polarity of the sheaf, in which every line 
corresponds to a plane to which it is perpendicular [cf. 7 1]. 
The fundamental conic of this polarity is imaginary, since 
there are no real lines in the sheaf, which lie on the corre 
sponding perpendicular plane. It can easily be shown that 
it contains all the pairs of circular points, which belong to 
the different planes in space, and that it appears as the com 
mon section of all spheres. From this property the name 
of circle at infinity is given to this fundamental metrical 
entity in space. 

Cf., e. g. F. ENRIQUES, Lezioni di Geometria proiettiva, 2 a. Ed. 
p. 177 188 (Bologna, Zanichelli, 1904). There is a German 
translation of the first edition of this work by H. FLEISCHER 
(Leipzig, 1903). 

Cayley s Absolute. \^j 

80. The two following questions naturally arise at 
this stage: 

(i) Can projective geometry be founded upon the Non- 
Euclidean hypothesis? 

(ii) If such a foundation is possible, can the metrical 
properties, as in the Euclidean case, be subordinated to the 

To both these questions the reply is in the affirmative. 
If RIEMANN S system is valid in space, the foundation of 
projective geometry does not offer any difficulty, since those 
graphical properties are immediately verified, which give rise 
to the ordinary projective geometry, after the improper entities 
are introduced. If the system of LOBATSCHEWSKY-BOLYAI is 
valid in space, we can also again lay the foundation of the 
projective geometry, by introducing, with suitable conventions, 
improper or ideal points, lines and planes. This extension will 
follow the same lines as were taken in the Euclidean case, in 
completing space with the elements at infinity. It would be 
sufficient, for this, to consider along with the proper sheaf 
(the set of lines passing through a point), two improper 
sheaves, one formed by all the lines which are parallel to a 
given line in one direction, the other by all the lines perpen 
dicular to a given plane; also to introduce improper points, 
to be regarded as the vertices of these sheaves. 

Even if the improper points of a plane cannot in this 
case, as in the Euclidean, be assigned to a straight line \thc 
line at infinity], yet they form a complete region, separated 
from the region of ordinary points (proper points) by a conic 
[limiting conic, or conic at infinity]. This conic is the locus 
of the improper points determined by the pencils of parallel 

In space the improper points are separated from the 
proper points by a non-ruled quadric [limiting quadric or 

Itj8 V. The Later Development of Non-Euclidean Geometry. 

quadric at infinity\ which is the locus of the improper points 
determined by sets of parallel lines. 

The validity of projective geometry having been estab 
lished on the Non-Euclidean hypotheses [KLEIN x ], to obtain 
the subordination of the metrical geometry to the projective 
it is sufficient to consider, as in the Euclidean case, the 
fundamental metrical entities (the absolute), and to interpret the 
metrical properties of figures as graphical relations between 
them and these entities. On the plane of LOBATSCHEWSKY- 
BOLYAI the fundamental metrical entity is the limiting conic, 
which separates the region of proper points from that of 
improper points, on the plane of RIEMANN it is an imaginary 
conic, defined by the absolute polarity of the plane [cf. p. 144]. 

In the one case as well as in the other, the metrical 
properties of figures are all the graphical properties which 
remain unaltered in the projective transformations 2 leaving the 
absolute fixed. 

These projective transformations constitute the 00^ dis 
placements of the Non-Euclidean plane. 

In the Euclidean case the said transformations, (which 
leave the absolute unaltered), are the oo 4 transformations of 
similarity, among which, as a special case, are to be found 
the 003 displacements. 

In space the subordination of the metrical to the pro- 

1 The question of the independence of Projective Geometry 
from the theory of parallels is touched upon lightly byj. KLEIN in 
his first memoir: Uber die sogenannle Nicht-Euklidische Geometric, 
Math. Ann. Bd. IV, p. 573 625 (1871). He gives a fuller treatment 
of the question in Math. Ann. Bd. VI, p. 112145 ( l %73)- This 
question is discussed at length in our Appendix IV p. 227. 

2 By the term projective transformation is understood such a 
transformation as causes a point to correspond to a point, a line to 
a line, and a point and a line through it, to a point and a line 
through it. 

Metrical Properties as Graphical. 

jective geometry is carried out by means of the limiting 
quadric (the absolute of space). If this is real, we obtain the 
geometry of LOBATSCHEWSKY-BOLYAI; if it is imaginary, we 
obtain RIEMANN S elliptic type. 

The metrical properties of figures are therefore the graph 
ical properties of space in relation to its absolute; that is, the 
graphical properties which remain unaltered in all the project- 
ive transformations wJiich leave the absolute of space fixed. 

81. How will the ideas of distance and of angle be 
expressed with reference to the absolute? 

Take a system of homogeneous coordinates (x I , x 2 , x^) 
on the projective plane. By their means the straight line is 
represented by a linear equation, and the equation of the 
absolute takes the form : 

Q xx = Ztfy- Xf Xj = O. 

Then the distance between two points X (x I} x 2 , x 3 ), 
Y(yi,y 2 ,y^ is expressed, omitting a constant factor, by the 
logarithm of the anharmonic ratio of the range consisting of 
X^ y, and the points M, N, in which the line X Y meets the 

If we then put 

and remember, from analytical geometry, that the anharm 
onic ratio of the four points X y Y, M, N is given by 

a,, i Q~a, 

the expression for the distance D xy will be : 

Introducing the inverse circular and hyperbolic functions, 

l6o V. The Later Development of Non- Euclidean Geometry. 

The constant >, which appears in these formulae, is 
connected with RIEMANN S Curvature K by the equation 


Similar considerations lead to the projective interpret 
ation of the conception of angle. The angle between two 
lines is proportional to the logarithm of the anharmonic ratio 
of the pencil which they form with the tangents from their 
point of intersection to the absolute. 

If we wish the measure of the complete pencil to be 
2 IT, as in the ordinary measurement, we must take the 
fraction 1:2* as the constant multiplier. Then to express 
analytically the angle between two lines u (u t , u 2J 3 ), 
v (z/ x , z/ 2 , z> 3 ), we put 

= X bij u r - Uj . 

If bij is the co factor of the element a,y in the dis 
criminant of Q xx , the tangential equation of the absolute is 
given by 

Vnu - O, 

and the angle between the two lines by the following 


,z/= -jlog 

* * 

Formulae for the Angle. 


^ ^ 


? cosh- 

< , z; ~ sin - 1 Y< z/z/ " ~_?f* v 

i/V v 

MM T z/z/ 
l/XJ/2 \j/ \lf 

I . , _ T uv ~ uu v 

<- u v = _ - 

(3 ) 

Similar expressions hold for the distance between two 
points and the angle between two planes, in the geometry 
of space. We need only suppose that 

Q = o, Y = o, 

represent the equations (in point and tangential coordinates) 
of the absolute of space, instead of the absolute of the plane. 

According as Q* x = O is the equation of a real quadric, 
without generating lines, or of an imaginary quadric, the 
formulae will refer to the geometry of LOBATSCHEWKY-BOLYAI, 
or that of RIEMANN. X 

82. The preceding formulae, concerning the angles 
between two lines or planes, contain those of ordinary 
geometry as a special case. Indeed if, for simplicity, we take 
the case of the plane, and the system of orthogonal axes, 
the tangential equation of the Euclidean absolute (the circular 
points, 8 79) is 

*i 2 + * 2 2 = O. 

The formula (2 ), when we insert 

Y = u, 2 + w 2 2 , Y r ,, = v* + z/ 2 2 , Y y = u# t + u 2 v 2 , 

1 For a full discussion of the subject of this and the pre 
ceding sections, see CLEBSCH-LINDEMANN, V^orlesungen ubcr Geometrie, 
Bd. II. Th. I, p. 461 et seq. (Leipzig, 1891). 


1 62 V. The Later Development of Non-Euclidean Geometry. 

U J V 1 + U 2 V 2 

<3u,v=* cos - 1 f 


from which we have 

/ v Ui 

cos (u, v) = . __ 

But the direction cosines of the line u ( u 2) 3 ) are 
cos (u, x) = -====.-, cos (uy) = 1=== =-, 

so that this equation can be written 

cos (u,v) = /!/ 2 -t- fn^m 2 

the ordinary expression for the angle between the two lines 
(A /,) and (/ 2 w 2 ). 

For the distance between two points the argument does 
not proceed so simply, when the absolute degenerates into 
the circular points. Indeed the points M", IV, where the line 
XY intersects the absolute, coincide in the point at infinity on 
this line, and the formula (i) gives in every case: 

^ = log (M^N^XY) = log i = o. 

However, by a simple artifice we can obtain the 
ordinary formula for the distance as the limiting case of 
formula (3). 

To do this more easily, let us suppose the equations 
of the absolute (not degenerate), in point and line coor 
dinates, reduced to the form : 

Q^. v = e^ 2 4. e# 2 2 + x 3 2 = o, 
Y = u, 2 + u 2 2 + e*3 2 = o. 
Then, putting 

equation (3) of the preceding section gives 

Euclid s Geometry as a Limiting Case. 163 

Z) xy = ik sin - 1 Ke A . 

Let e be infinitesimal. Omitting terms of a higher 
order, we can substitute V eA for sin"" 1 V A in this formula 
If we now choose k 2 infinitely large, so that the product 
ik V*. remains finite and equal to unity for every value of e, 
the said formula becomes 

+ txS + x tyS + + 
Let e now tend to the limit zero. The tangential 
equation of the absolute becomes 

uS + u 2 * = o; 

and the conic degenerates into two imaginary conjugate 
points on the line u 3 = o. The formula for the distance, 
on putting 

x . xi V- yi 
Xl ~ *3 l ~^ 
takes the form 

D xy = (XYtf + (X 2 - 2 , 

which is the ordinary Euclidean formula. We have thus ob 
tained the required result. 

We note that to obtain the special Euclidean case from 
the general formula for the distance, we must let k 2 tend to 
infinity. Since RIEMANN S curvature is given by -g , this 
affords a confirmation of the fact that RIEMANN S curvature 
is zero in Euclidean space. 

83. The properties of plane figures with respect to a 
conic, and those of space with respect to a quadric, together 
constitute projective metrical geometry. This was first studied 
by CAYLEY, X apart from its connection with the Non-Euclid- 

i Sixth Memoir upon Qitantics. Phil. Trans. Vol. CXLIX, p. 6 1 
90 (1859). Also Collected Works, Vol. II, p. 561592. 


164 V * T1 *e Later Development of Non-Euclidean Geometry. 

can geometries. These last relations were discovered and 
explained some years later by F. KLEIN. x 

To KLEIN is also due a widely used nomenclature for 
the projective metrical geometries. He gives the name hyper 
bolic geometry to CAYLEY S geometry, when the absolute is 
real and not degenerate: elliptic geometry, to that in which 
the absolute is imaginary and not degenerate: parabolic 
geometry, to the limiting case of these two. Thus, in the 
remaining articles, we can use this nomenclature to describe 
the three geometrical systems of LOBATSCHEWSKY-BOLYAI, of 
RIEMANN (elliptic type), and of EUCLID. 

Representation of the Geometry of Lobatschewsky- 
Bolyai on the Euclidean Plane. 

84. To the projective interpretation of the Non- 
Euclidean measurements, of which we have just spoken, may 
be added an interesting representation which can be given 
of the Hyperbolic Geometry on the Euclidean plane. To ob 
tain it, we take on the plane a real, not degenerate, conic : 
e. g. a circle. Then we make the following definitions, relative 
to this circle : 

Plane = region of points within the circle. 

Point = point inside the circle. 

Straight line = chord of the circle. 

We can now easily verify that the postulate that a 
straight line is determined by two points, and the postulates 
regarding the properties of straight lines and angles, can be 
expressed as relations, which are always valid, when the above 
interpretations are given to these terms. 

But in the further development of this geometry we add 

1 Cf. Uber die sogenannte Nicht-Euklidische Geometric. Math. 
Ann. Bd. IV, p. 573625 (1871). 

Representation on the Euclidean Plane. 165 

to these the postulates of congruence, contained in the 
following principle of displacement. 

If we are given two points A, A on the plane, and the 
straight lines a, a, respectively passing through them, there 
are four methods of superposing the plane on itself, so that 
A and a coincide respectively with A and a . More precisely: 
one method of superposition is denned by taking as corre 
sponding to each other, one ray of a and one ray of a, one 
section of the plane bounded by a and one section bounded 
by a . Two of these displacements are direct congruences 
and two converse congruences. 

With the preceding interpretations of the entities, point, 
tin.- and plane, the principle here expressed is translated 
into the following proposition: 

If a conic (e.g., a circle) is given in a plane, and two 
internal points A, A are taken, as also two chords a, a , re 
spectively passing through them, there are four protective trans 
formations of the plane, which change into itself the space 
within the conic, and which make A and a correspond respect 
ively to A and a . 

To fix one of them, it is sufficient to make sure that a 
given extremity of a corresponds to a given extremity of a, 
and that to one section of the plane bounded by a, cor 
responds a definite section of the plane bounded by a . Of 
these four transformations, two determine on the conic a 
projective correspondence in the same sense, and two a pro- 
icctive correspondence in the opposite sense. 

85. We shall prove this proposition, taking for sim 
plicity two distinct conies T, T , in the same plane or other 

Let M, N be the extremities of the chord a [cf. Fig. 58]. 
Also M , N those of a [cf. Fig. 59]. 

1 66 v - The Later Development of Non-Euclidean Geometry. 

Let -P, P be the poles of a, a with respect to the two 

On this understanding, the line PA intersects the 
conic T in two real and distinct points R, S\ also the line 
P A intersects the conic T in two real and distinct points 

A projective transformation which changes T into T , the 
line a into a, and the point A into A t will make the point P 
correspond to P , and the line PA to the line P A . 





Fig. 59. 

Thus this transformation determines a projective cor 
respondence between the points of the two conies, in which 
the pair of points M , W corresponds to the pair of points 
M, N\ and the pair of points R , S to R, S. 

Vice versa, a projective transformation between the two 
conies, which enjoys this property, is associated with a pro 
jective transformation of the two planes, such as is here de 
scribed. 1 

But if we consider the two conies T, T , we see that to 

1 For this proof, and the theorems of Projective Geometry 
upon which it is founded, see Chapter X, p. 251 253 of the work 
of ENRIQUES referred to on p. 156. 

Projective Transformations. 1 67 

the points of the range MNRS on T may be made to cor 
respond the points of any one of the following ranges on T : 
M N R S 
N M S R 
M N S K 
N M R S . 

In this way we prove the existence of the four project- 
ive transformations of which we have spoken in the propos 
ition just enunciated. 

If we suppose that the two conies coincide, we do not 
need to change the 
preceding argument in 
any way. We add, how- p 
ever, that of the four 
transformations only 
one makes the segment 
AM correspond to the 
segment A M\ if at the 
same time the shaded 
parts of the figure cor 
respond to each other. 

Further the two transformations defined by the ranges 
/ MNRS \ / MNRS \ 
V M N R S /, V N M S R ) 

determine projections in the same sense, while the other two, 
defined by the ranges : 

/ MNRS \ / MNRS \ 
V M N S R ) V N M R S ) 
determine projections in the opposite sense. 

86. With these remarks, we now return to complete 
the definitions of 8 84, relative to a circle given on the 

Plane = region of points within the circle. 

1 68 V. The Later Development of Non-Euclidean Geometry. 

Point = point within the circle. 

Straight Line = chord of the circle. 

Displacements = projective transformations of the plane 
which change the space within the circle into itself. 

Semi- Revolutions = homographic transformations of the 

Congruent Figures = figures which can be transformed 
the one into the other by means of the projective trans 
formations named above. 

The preceding arguments permit us to affirm at once 
that all the propositions of elementary plane geometry, asso 
ciated with the concepts straight line, angle and congruence, 
can be readily translated into properties relative to the 
system of points inside the circle, which we denote by (S). 
In particular let us see what corresponds in (S) to two per 
pendicular lines in the ordinary plane. 

To this end we note that if r, s are two perpendicular 
lines, a semi-revolution of the plane about s will superpose 
r upon itself, exchanging, however, the two rays in which it 
is divided by s. 

According to the above definitions, a semi-revolution in 
(S) is a homographic transformation, which has for axis a 
chord s of the circle and for centre the pole of the chord. 
The lines which are unchanged in this transformation, in ad 
dition to s, are the lines passing through its centre. Thus 
in the system (S) we must call two lines perpendicular, when 
they are cojijugate with respect to the fundamental circle. 

We could easily verify in (S) all the propositions on 
perpendicular lines. In particular, that if we draw the (imag 
inary) tangents to the fundamental circle from the common 
point of two conjugate chords in (S}, these tangents form 
a harmonic pencil with the perpendicular lines [cf. p. I55]. 1 

1 This representation of the Non-Euclidean plane has been 

The Distance between two Points. 169 

87. Let us now see how the distance between two 
points can be expressed in this conventional measurement, 
which is being taken for the interior of the circle. 

To this end we introduce a system of orthogonal coord 
inates (x, y), with origin at the centre of the circle. 

The distance between two points A (x, y), B (x f , y) 
in the plane with which we are dealing cannot be represen 
ted by the usual formula 

since it is not invariant for the projective transformations 
which we have called displacements. The distance must be a 
function of the coordinates, invariant for the said transforma 
tions, which for points on the straight line possesses the dis 
tributive property given by the formula 

dist. (AB) = dist. (AC) + dist. (CB). 

Now the anharmonic ratio of the four points A, -B, M, 
N, where M, N are the extremities of the chord AB } is a 
relation between the coordinates (x, y}, (x, y ) of AB, 
remaining invariant for all projective transformations which 
leave A the t fundamental circle fixed. The most general ex 
pression, possessing this invariant property, will be an arbi 
trary function of this anharmonic ratio. 

If we remember that the said function must be distrib 
utive in the sense above indicated, we must assume that, 
except for a multiplier, it is equal to the logarithm of the 
anharmonic ratio, 

We shall thus have 

distance (AB) = |- log (ABMN). 

employed by GROSSMANN in carrying out a number of the con 
structions of Non-Euclidean Geometry. Cf. Appendix, III, p. 225. 

T7O V. The Later Development of Non-Euclidean Geometry. 

In a similar way we proceed to find the proper ex 
pression for the angle between two straight lines. In this case 
we must notice that if we wish the right angle to be ex 
pressed by , we must take as constant multiplier of the 
logarithm the factor i : 21. 

Then we shall have for the angle between a and , 
< a, b = - 2 . log (abmii), 

where w, n are the conjugate imaginary tangents from the 
vertex of the angle to the circle, and (a b m n) is the an- 
harmonic ratio of the four lines a, b, m and , expressed 
analytically by 

sin (a m} sin (a 11} 
sin (b m} sin (b n} 

88. A glance at what was said above on the sub 
ordination of the metrical to the projective geometry ( 81) 
will show clearly that the preceding formulas, regarding the 
distance and angle, agree with those which we would have in 
the Non-Euclidean plane, if the absolute were a circle. This 
would be sufficient to suggest that the geometry of the system 
(S) gives a concrete representation of the geometry of 
LoBATSCHEWSKY-BoLYAi. However, as we wish to discuss 
this point more fully, let us see how the definition and pro 
perty of parallels are translated in (S). 

Let r (U-L, u 2 , z* 3 ) and r (v lt v 2 , z> 3 ) be two different 
chords of the fundamental circle. 

Let the circle be referred to an orthogonal Cartesian 
set of axes, with the centre for origin, and let us take the 
radius as unit of length. 

Then we have 

x 2 +y* 1=0, 
u 2 + v 2 i=0, 
for the point and line equation of the circle. 

The Angle between two Lines. \ f i\ 

Making these equations homogeneous, we obtain 

*i 2 + * a a * 3 2 = 0, 

v t * + U2* 7/ 3 2 = O. 

The angle < r, r between the two straight lines r and 
r can be calculated by means of the formula (3 ) of 81, 
if we put 

Ywz, = fc^ + U 2 V 2 Uf) y 

We thus obtain 

sn -C r r 

But the lines r ; r are given by 

x I u I + x 2 u 2 ^-x^ 3 = O, 
X& l + x 2 v 2 +x 3 v 3 = O; 
and they meet in the point, 

Thus the preceding expression for this angle takes 
the form 

, x 

(4) sin 

From this it is evident that the necessary and sufficient 
condition that the angle be zero is that the numerator of 
this fraction should vanish. 

Now if this numerator is zero, the point (x lt x 2) x 3 ), in 
which the chords intersect, must lie on the circumference of 
the fundamental circle, and vice versa (Fig. 61). 

Therefore in our interpretation of the geometrical pro 
positions fy means of the system (S), we must call two chords 
parallel, when they meet in a point on the circumference of the 

172 V. The Later Development of Non-Euclidean Geometry. 

fundamental circle, since the angle between those two chords 
is zero. 

Since there are two chords through any point within a 
circle which join this point to the ends of any given chord, 
the fundamental proposition of hyperbolic geometry will be 
verified for the system (S). 

89. We proceed to find for the system (S) the 
formula regarding the angle of parallelism. To do this we 
first calculate the angle OMN, between the axis of y and 
the line MN, joining a point M on the axis ofy to the ex 
tremity of the axis of x (Fig. 62). 

Fig. 6x. 

Fig. 62. 

Denoting by a the ordinary distance of the two points 
M and (9, the homogeneous coordinates of the line MN and 
the line <9J/are, respectively (a } i, a], (i, o, o) and the 
coordinates of their common point are (o, a, i). 

Then from (4) of the preceding article, 
sin <^ OMN = yi-a*. 

On the other hand, the distance, according to our con 
vention, between the two points O and M is given by (2) of 

8 1 as 


OM = k cosh - 1 




The Angle of Parallelism. 

Comparing these two results, we have 
OM i 

a relation which agrees with that given by TAURINUS, Lo- 
BATSCHEWSKY and BOLYAI for the angle of parallelism [cf. 
p. 90]. 

90. We proceed, finally, to see how the distance be 
tween two neighbouring points (the dement of distance) is 
expressed in the system (*S), so that we may be able to 
compare this representation of the hyperbolic geometry with 
that given by BELTRAMI [cf. 69]. 

Let (#, y\ (x + dx, y-\-dy) be two neighbouring points. 
Their distance ds is calculated by means of (2) of 81 if we 
substitute : 

Since the angle is small, we may substitute the sine for 
the angle, and we have 

= (*** + <) (i - ^ 2 y 2 ) + (*** +y<*y) 2 

( X 2 +y2 _ J) (( X + dx y + (y + d yY _ j)) 

Thus, omitting terms higher than the second order, 
we have 


Now we recall that BELTRAMI, in 1868, interpreted the 
geometry of LOBATSCHEWSKY-BOLYAI by that on the surfaces 
of constant negative curvature. The study of the geometry 
on such surfaces depends upon the use of a system of coord 
inates on the surface, and the law according to which the 
element of distance (ds} is measured. The choice of a suitable 

1/4 ^ ^k ft Later Development of Non-Euclidean Geometry. 

system (u,i>) enabled BELTRAMI to put the square of ds in 
this form: 

, 2 -f- 


where the constant k 2 is the reciprocal, with its sign changed, 
of the curvature of the surface. 1 

In studying the properties of these surfaces and in mak 
ing a comparison between them and the metrical results of 
the geometry of LOBATSCHEWSKY-BOLYAI, BELTRAMI in his 
classical memoir, quoted on p. 138, employed the following 
artifice : 

He represented the points of the surface on an aux 
iliary plane, such that the point (u, v) of the surface corre 
sponded to the point on the plane whose Cartesian coord 
inates (x, y} were (u, v). The points on the surface were 
then represented by points inside the circle 

x 2 +y 2 i = O; 

the points at infinity on the surface by points on the cir 
cumference of the circle: its geodesies by chords: parallel 
geodesies by chords meeting in a point on the circumference 
of the said circle. Then the expression for (ds} 2 took the 
same form as that given in (5), which states the form to be 
used for the element of distance in the system (S). 

It follows that, by his representation of the surfaces of 
constant negative curvature on a plane, BELTRAMI was 
led to one of the projective metrical geometries of CAYLEY, 
and precisely to the metrical geometry relative to a funda 
mental circle, given above in 80, 81. 

1 Risoluzionc del problema di riportarc i punti di una superficie 
sopra un piano in modo che le linee geodetiche vengano rapprtsentate 
da linee retle. Ann. di Mat. T. VII, p. 185 204 (1866). Also 
Opere Matematiche. T. I, p. 262280 (Milan, 1902). 

Beltrami s Geometry and Projective Geometry. j^c 

91. The representation of plane hyperbolical geo 
metry on the Euclidean plane is capable of being extended to 
the case of solid geometry. To represent the solid geometry 
of LoBATSCHEWSKY-BoLYAi in ordinary space we need only 
adopt the following definitions for the latter: 

Space = Region of points inside a sphere. 

Point = Point inside the sphere. 

Straight Line = Chord of the sphere. 

Plane = Points of a plane of section which are inside 
the sphere. 

Displacements = Projective transformations of space, 
which change the- region of the points inside the 
sphere into itself, etc. 

With this Dictionary the propositions of hyperbolic 
solid geometry can be translated into corresponding proper 
ties of the Euclidean space, relative to the system of points 
inside the sphere. 1 

Representation of Riemann s Elliptic Geometry in 
Euclidean Space. 

92. So far as regards plane geometry, we have already 
remarked [pp. 142 3] that the geometry of the ordinary 
sheaf of lines gives a concrete interpretation of the elliptical 
system of RIEMANN. Therefore, if we cut the sheaf by an 
ordinary plane, completed by the line at infinity, we obtain 
a representation on the Euclidean plane of the said RIE 
MANN S plane. 

1 BELTRAMI considers the interpretation of Non-Euclidean Solid 
Geometry, and, in general, of the geometries of manifolds of 
higher order in space of constant curvature, in his memoir: Teoria 
fondamentale degh spazii di curvatura costante. Ann. di Mat. (2), 
T. II, p. 232255 (1868). Opere Mat. T. I, p. 406429 (Milan, 

176 V. The Later Development of Non-Euclidean Geometry. 

If we wish a representation of the elliptic space in the 
Euclidean space, we need only assume in this a single-valued 
polarity, to which corresponds an imaginary quadric, not 
degenerate. We must then take, with respect to this quadric, 
a system of definitions analogous to those indicated above 
in the hyperbolic case. We do not pursue this point further, 
as it offers no fresh difficulty. 

However we remark that in this representation all the 
points of the Euclidean space, including the points on the plane 
at infinity, would have a one-one correspondence with the points 
of Riemann s space. 

Foundation of Geometry upon Descriptive 

93. The principles explained in the preceding sections 
lead to a new order of ideas in which the descriptive propert 
ies appear as the first foundations of geometry, instead of 
congruence and displacement, of which RIEMANN and HELM- 
HOLTZ availed themselves. We note that, if we do not wish 
to introduce at the beginning any hypothesis on the inter 
section of coplanar straight lines, we must start from a 
suitable system of postulates, valid in a bounded region of 
space, and that we must complete the initial region later by 
means of improper points, lines and planes [cf. p. 157].* 

When projective geometry has been developed, the 
metrical properties can be introduced into space, by adding 
to the initial postulates those referring to displacement or 

1 For such developments, cf. KLEIN, loc. cit. p. 158: PASCH, 
Vorlesungen uber neuere Geometric, (Leipzig, 1882)} SCHUR, Uber die 
Einfuhrung der sogenannten idea! en Elemente in die projective Geometrie, 
Math. Ann. Bd. XXXIX, p. 113124 (1891): BONOLA, Sulla intro- 
duzione degli elementi improprii in geometria proietli ua y Giornale di 
Mat. T. XXXVIII, p. 105116 (1900). 

Foundation of Geometry upon Descriptive Properties. \ f j>i 

congruence. By so doing we find that a certain polarity of 
space, allied to the metrical conceptions, becomes trans 
formed into itself by all displacements. Then it is shown 
that the fundamental quadric of this polarity can only be: 

a) A real, non-ruled quadric ; 

b} An imaginary quadric (with real equation); 

c) A degenerate quadric. 

Thus the three geometrical systems, which RIEMANN and 
HELMHOLTZ reached from the conception of the element of 
distance, are to be found also in this way. 1 

The Impossibility of proving Euclid s Postulate. 

94. Before we bring to a close this historical treat 
ment of our subject it seems advisable to say a few words 
on the impossibility of demonstrating Euclid s Postulate. 

The very fact that the innumerable attempts made to 
obtain a proof did not lead to the wished-for result, would 
suggest the thought that its demonstration is impossible. In 
deed our geometrical instinct seems to afford us evidence 
that a proposition, seemingly so simple, if it is provable, 
ought to be proved by an argument of equal simplicity. But 
such considerations cannot be held to afford a proof of the 
impossibility in question. 

If we put EUCLID S Postulate aside, following the devel 
opments of GAUSS, LOBATSCHEWSKY and BOLYAI, we can 
construct a geometrical system in which no contradictions 
are met. This seems to prove the logical possibility of the 
Non-Euclidean hypothesis, and that EUCLID S Postulate is 
independent of the first principles of geometry and therefore 
cannot be demonstrated. However the fact that contradictions 

1 For the proof of this result see BONOLA., Determinazione 
per -via geometrica dei ire tipi de spazio ; iperbolico, parabolico, ellitiico. 
Rend. Circ. Mat. Palermo, T. XV, p. 5665 (1901). 


jy3 V. The Later Development of Non-Euclidean Geometry. 

have not been met is not sufficient to prove this; we must 
be certain that, proceeding on the same lines, such con 
tradictions could never be met. This conviction can be 
gained with absolute certainty from the consideration of the 
formulae of Non-Euclidean geometry. If we take the system 
of all the sets of three numbers (x, y, z), and agree to con 
sider each set as an analytical point, we can define the 
distance between two such analytical points by the formulae 
of the said Non-Euclidean Trigonometry. In this way we 
construct an analytical system, which offers a conventional 
interpretation of the Non-Euclidean geometry, and thus 
demonstrates its logical possibility. 

In this sense the formulae of the Non- Euclidean Trigon 
ometry of Lobatschewsky-Bolyai give the proof of the independ 
ence of Euclid s Postulate from the first principles of geometry 
(regarding the straight line, the plane and congruence). 

We can seek a geometrical proof of the said independ 
ence, on the lines of the later developments of which we 
have given an account. For this it is necessary to start from 
the principle that the conceptions, derived from our intu 
ition, independently of the correspondence which they find 
in the external world, are a priori logically possible-, and that 
thus the Euclidean geometry is logically possible and every 
set of deductions founded upon it. 

But the interpretation which the Non-Euclidean plane 
hyperbolic geometry finds in the geometry on the surfaces 
of constant negative curvature, offers, up to a certain point, 
a first proof of the impossibility of demonstrating the Eu 
clidean postulate. To put the matter in more exact terms: 
by this means it is established that the said postulate cannot 
be demonstrated on the foundation of the first principles of 
geometry, held valid in a bounded region of the plane. In 
fact, every contradiction, which would arise from the 
other postulate, would be translated into a contradiction 

Euclid s Postulate cannot be Proved. 

in the geometry on the surfaces of constant negative curv 

However, since the comparison between the hyperbolic 
plane and the surfaces of constant negative curvature, exists, 
as we have seen, only for bounded regions, we have not thus 
excluded the possibility that the Euclidean postulate might 
be proved for the complete plane. 

To remove this uncertainty, it would be necessary to 
refer to the abstract manifold of constant curvature, since no 
concrete surface exists in ordinary space, in which the com 
plete hyperbolic geometry holds [cf. 73]. 

But, even so, the impossibility of proving Euclid s Pos 
tulate would have been shown only for plane geometry. There 
would still remain the question of the possibility of proving 
it by means of the considerations of solid geometry. 

The foundation of geometry, on RIEMANN S principles, 
whereby the ideas of the geometry on a surface are extended 
to a three-dimensional region, gives the complete proof of the 
impossibility of this demonstration. This proof depends on 
the existence of a Non-Euclidean analytical system. Thus we 
are brought to another analytical proof. The same remark 
applies also to the investigations of HELMHOLTZ and LIE, 
though it might be argued that the latter also offer a geomet 
rical proof, from the existence of transformation groups of 
the Euclidean space, similar to the groups of displacements of 
the Non-Euclidean geometry. Of course, it must be under 
stood that we here consider geometry in its fullest sense. 

But the proof of the impossibility of demonstrating Eu 
clid s Postulate, which is based upon the protective measure 
ments of Cayley, is simpler and easier to follow geometrically. 

This proof depends upon the representation of the 
Non-Euclidean geometry by the conventional measurement 
relative to a circle or to a sphere, an interpretation which we 


I So V. The Later Development of Non-Euclidean Geometry. 

have developed at length in the case of the plane [ 84 


Further the proof of the logical possibility of RIEMANN S 

elliptic hypothesis can be just as easily derived from these 
projective measurements. For the plane, the interpretation 
which we have given of it as the geometry of the sheat 
will be sufficient [ yi]. x 

1 Another neat and simple proof of the independence of the 
Fifth Postulate is to be found in the representation of the Non- 
Euclidean plane, employed by KLEIN and POINCARE. In this the 
points of the Non-Euclidean plane appear as points of the upper 
portion of the Euclidean plane, and the straight lines of the Non- 
Euclidean plane as semicircles, perpendicular to the straight bound 
ary of this halfplane; etc. The Elliptic Geometry can be repres 
ented in a similar way; and the Hyperbolic and Elliptic Solid 
Geometries can also be brought into correspondence with the 
Euclidean Space. An account of these representations is to be 
found in WEBER und WELLSTEIN S Encyklopddie der Elementar- 
Mathematik, Bd. II $ 9 n P- 39 Si (Leipzig, 1905) and in 
Chapter II of the Nicht-Euklidische Geometrie by H. LIEBMANN 
(Sammlung SCHUBERT, 49, Leipzig, 1905). 

In Appendix V of this volume a similar argument is given, 
based upon the discussion in WEBER- WELLSTEIN S volume. Points 
upon the Non-Euclidean plane are represented by pairs of points 
inverse to a fixed circle on the Euclidean plane; and straight 
lines upon the one, are circles orthogonal to the fixed circle on 
the other. 

Appendix I. 

The Fundamental Principles of Statics and 
Euclid s Postulate. 

On the Principle of the Lever. 

i. To demonstrate the Principle of the Lever, ARCHI 
MEDES [287 212] avails himself of several hypotheses, some 
expressed and others implied. Among the hypotheses 
passed over in silence, in addition to that which we would 
now call the hypothesis of increased constraint*, there is one 
which definitely concerns the equilibrium of the lever, and 
can be expressed as follows: 

When a lever is suspended from its middle point, it is in 
equilibrium, if a weight iP is applied at one end, and at the 
other another lever is hu?ig by its middle point, each of its ends 
supporti?ig a weight P. 2 

We shall not discuss the various criticisms upon ARCHI 
MEDES use of this hypothesis, nor the different attempts made 
to prove it. 3 In this connection we shall refer only to the 

1 This hypothesis can be enunciated as follows: If several bodies, 
subjected to constraints, are in equilibrium under the action of given 

forces, they will still be in equilibrium, if new constraints are added 
to those already in existence. Cf., for example, J. ANDRADE, Lemons 
de Mecanique Physique, p. 59 (Paris, 1898). 

2 Cf. Archimedis opera omnia : critical edition by J. L. HEIBERG ; 
Bd. II, p. 142 et seq. (Leipzig, 1881). 

3 Cf., for example, E. MACH, Die Mechanik in ihrer Ent- 

1 82 Appendix I. The Fundamental Principles of Statics etc. 

arguments of LAGRANGE, since these will show, clearly and 
simply, the important link between this hypothesis and the 
Parallel Postulate. 

2. Let ABD be an isosceles triangle (AD = BD), 
from whose angular points A and B are suspended two 
(cf. Fig. 63) equal weights P, while a weight equal to 2? is 

suspended from D. 

This triangle will be in equilibrium 

about the straight line MN, joining 
the middle points of the equal sides, 
since each of these sides may be 
regarded as a lever from whose ex 
tremities equal weights are hung. 

But the equilibrium of the figure 
will also be secured, if the triangle 
rests upon a line passing through 
the vertex D and the middle point 
C of the side AB. Therefore, if E 
is the common point of CD and MN t 
the triangle will be in equilibrium, when suspended from E. 

Or , continues LAGRANGE, comme 1 axe \MN\ passe 
par le milieu des deux cotes du triangle, il passera aussi 
necessairement par le milieu de la droite menee du sommet 
du triangle au milieu [C] de sa base; done le levier trans 
versal [CD] aura le point d appui [] dans le milieu et 
devra, par consequent, etre charge egalement aux bouts 
[C, D\ : done la charge que supporte le point d appui du 
levier, qui fait la base du triangle, et qui est charge, a ses 

wickelung, (3. Aufl., Leipzig, 1897); English translation by T. J. Mc- 
CORMACK (Open Court Publishing Co. Chicago, 1902). Also, for 
the different hypotheses from which the proof of the principle of 
the lever, can be obtained, see P. DUHEM, Les origins de la stati- 
que, [Paris, 1905), especially Appendix C, Sur les divers axiomes 
d ou se petti deduire la t hear if du levier. 

Statical Hypothesis equivalent to Postulate V. j37 

deux extremites de poids egaux, sera egale au poids double 
du sommet et, par consequent, egale a la somme des deux 
poids. x 

3. LAGRANGE S argument contains implicitly some 
hypothecs of a statical nattire, regarding symmetry, addition 
of constraints, 2 etc.; and, in addition, it involves a geometrical 
property of the Euclidean triangle. But if we wish to omit 
the latter, a course which for certain reasons seems natural, 
the preceding conclusions will be modified. 

Indeed, though we may still assume that the triangle 
ABD is in equilibrium about the point E, where the lines 
MN and CD intersect, we cannot assert that E is the middle 
point of C>, as this would be equivalent to assuming 
EUCLID S Postulate. Consequently, we cannot assert that the 
single weight 2 P, applied at C, can be substituted for the two 
weights at A and B, since, if such a change could take place, 
a lever would be in equilibrium, with equal weights at its ends, 
about a point which cannot be its middle point. 

Vice versa, if we assume, with ARCHIMEDES, that two 
equal weights at the end can be replaced by a double 
weight at the middle point of the lever, then we can easily 
deduce that E is the middle point of CD, and from this it 
will follow that ABD is a Euclidean triangle. 

Hence we have established the equivalence of Euclid s 
Fifth Postulate and the said hypothesis of Archimedes. Such 
equivalence is, of course, relative to the system of hypotheses 
which comprises, on the one hand, the above-named statical 
hypotheses, and, on the other, the ordinary geometrical 

1 Oeuvres de Lagrange, T. XI, p. 4 5. 

2 For an analysis of the physical principles on which ordinary 
statics is founded, cf. F. ENRIQUES, Problemi delta Sdenza. Cap. V. 
(Bologna, 1906). German translation, (Leipzig, 1910). 

1 84 Appendix I. The Fundamental Principles of Statics etc. 

With the modern notation, we can speak of forces, 
of the composition of forces, of resultants, instead of weights, 
levers, etc. 

Then the hypothesis referred to takes the following 

The resultant of two equal forces in the same plane, applied 
at right angles to the extremities of a straight line and towards 
the same side of it, is a single force at the middle point of the 
line, of double the intensity of the given forces. 

From what we have said above, if this law for the com 
position of forces were true, it would follow that the ord 
inary theory of parallels holds in space. 

On the Composition of Forces Acting at a Point. 

4. The other fundamental principle of statics, the 
law of the Parallelogram of Forces, from the usual geom 
etrical interpretation which it receives, is closely connected 
with the Euclidean nature of space. However, if we examine 
the essential part of this principle, namely, the analytical 
expression for the resultant R of two equal forces P, acting 
at a point, it is easy to show that it exists independently of any 
hypothesis on parallels. 

This can be made clear by deducing the formula 

R = 2 P cos a, 

where 2 a is the angle formed by the two concurrent forces 
from the following principles: 

1) Two or more forces, acting at the same point, have 
a definite resultant. 

2) The resultant of two equal and opposite forces 
is zero. 

3) The resultant of two or more forces, acting at a 
point, along the same straight line, is a force through the 
same point, equal to the sum of the given forces, and along 
the same line. 

Composition of Concurrent Forces. 

I8 5 

4) The resultant of two equal forces, acting at the same 
point, is directed along the line bisecting the angle between 
the two forces. 

5) The magnitude of the resultant is a continuous funct 
ion of the magnitude of the components. 

Let us see briefly how we establish our theorem. The 
value R of the resultant of two forces of equal magnitude P, 
enclosing the angle 2 a, is a function of P and a only. 

Thus we can write 

R- 2/GP,a). 

A first application of the principles named above shows 
that R is proportional to P, and this result is independent 
of any hypothesis on parallels [cf. note i, p. 195]. Thus the 
preceding equation can be written more simply as 

We now proceed to find the form of /(a). 

5. Let us calculate /"(a) for some particular value 
of the angle. 
(I) Let a = 45. 

At the point O at which act i p Q 

the two forces jP I} P 2i of equal "2 
magnitude P, let us imagine two 
equal and opposite forces applied, 
perpendicular to R and of magni 
tude y (cf. Fig. 64). 

At the same time let us imag 
ine R decomposed into two others, 
directed along R and of magni 
tude . 


Fig. 64. 

We can then regard each force P as the resultant of 


two forces at right angles, of magnitude 

1 86 Appendix I. The Fundamental Principles of Statics etc. 

We thus have 

P = 2 . 4 ./(4S). 

On the other hand, R being the resultant of P and P 2 , 
we have 

From these two equations we obtain 

7(45) = ~ VT, 

(II) Again let a = 60. 

In this case apply at O a force 1? equal and opposite 
to R (cf. Fig. 65). The system of the two forces P and of 
1? is in equilibrium. 

Thus by symmetry, R = P, 
Therefore, JR. = P. 
But, on the other hand, 

R=, 2 />/(6o). 

Therefore/ (60) = y. 

(IU) Again let a = 36. 

At O let the five forces P l , P 2 . . P s , of magnitude />, be 

Special Cases. 

applied, such that each of them forms with the next an angle 
of 72 (cf. Fig. 66). 

This system is in equilibrium. 

For the resultant R of P 2 and P 3 , we have 

R = 2/>7( 3 6). 

For the resultant R of P l and P 4 , we have 
R = 2/>7( 7 2). 

On the other hand, R has the same direction as P 5 ; 
that is, a direction opposite to that of R. 

Therefore 2/7(36) = 2^7(72) + P. 

(1) Therefore 27(36) 27(72) + i. 

If, instead, we take the resultants of P I and P 2 , and of 
jP 3 and PI, we obtain two forces of magnitude 2 P f (36), 
containing an angle of 144. 

Taking the resultant of these two, we obtain a new 
force R" of magnitude 

4 ^/(36 ) 7 (7 2). 

Now R", by the symmetry of the figure, has the same 
line of action as P 5 , but acts in the opposite direction. 
Thus, since equilibrium must exist, 

/>= 4 ^/(36) / (7 2). 

(2) Therefore i = 4 / (3 6) f (7 2). 
From the two equations (i) and (2) we obtain 

4 4 

on solving for f (3 6) and f (7 2). 

6. By arguments similar to those used in the pre 
ceding section we could deduce other values for f (a). 
However, if we restrict ourselves only to those just found, 

1 88 Appendix I. The Fundamental Principles of Statics etc. 

and compare them with the corresponding values of cos a, 
we obtain the following table : 

cos = i /(0) = i 

T _1_ *\/~T~ 

cos 36 = 

cos 45 




V . 

cos 60 = 

cos 72 


/( 9 ) = O. 

This table suggests the 
identity of the two functions 
/(a) and cos a. For fuller 
p confirmation of this fact, we 
determine the functional 
equation which / (a) satis- 
3 R 2 fies(cf. Fig. 67). 

To this end let us con 
sider four forces P It P 2 , 
P 3 , P^ of magnitude P, 
acting at one point, forming 
with each other the following angles 


V- 2((I-P) 

= 2 (a + p). 

We shall determine the resultant R of these four forces 
in two different ways. 

Taking P t with P 2 , and P 3 with /* 4 we obtain two forces 
./?i and ^? 2 , of magnitude 

The General Case. l8o 

inclined at an angle 2 p. Taking the resultant of R r and R z , 
we have a force R, such that 

On the other hand, taking P l with /> 4 , and .P 2 with P 3 , 
we obtain two resultants, both along the direction of R, and 
of magnitudes 

2Pf(a + p), 2/>/(a P), 


These two forces have a resultant equal to their sum, 
and thus 

/?= 2/V(a + P)+2-P/(a P). 
Comparing the two values of R, we find that 

(i) 2/(a)/(p) = /(a + p) + /(<*- P) 

is the functional equation required. 
If we now remember that 
cos (a + p) + cos (a P) = 2 cos a cos p, 

and take account of the identity between / (a) and cos a in 
the preceding table for certain values of a, and the hy 
pothesis that / (a) is continuous, without further argument 
we can write 

/(a) = cos a. 
It follows that 

R 2 P cos a. 

The validity of this formula of the Euclidean space is 
thus also established for the Non-Euclidean spaces. 

7. The law of composition of two equal concurrent 
forces leads to the solution of the general problem of the 
resultant, since we can assign, without any further hypothesis, 
the components of a force R along two rectangular axes 
through its point of application O. 

Appendix I. The Fundamental Principles of Statics etc. 

Let the two perpendicular lines be taken as the axes 
of x and y, and let R make the angles a, {3 with them 
(cf. Fig. 68). 

Y ft Through O draw the line 

which makes an angle a with 
Ox and an angle p with Oy. 
Imagine two equal and oppos 
ite forces P t and P 2 to act 
along this line at O, their mag- 


nitude being . Also imagine 
the force R replaced by the 
two equal forces P, of magni- 


tude , acting in the same 
direction as R. 

Then the system P lt P 2 , P, PhasR for resultant. But 
P! and P, taken together, have a resultant 

X = R cos a 
along Ox: and P 2 and P, taken together, have a resultant 

along Oy. 

These two forces are the components of R along the 
two perpendicular lines. As to their magnitudes, they are 
identical with what we would obtain in the ordinary theory 
founded upon the principle of the Parallelogram of Forces. 
However, the lines OX and O V t which represent the com 
ponents up on the axes, arenot necessarily the projections of R, 
as in the Euclidean case. Indeed we can easily see that, if 
these lines were the orthogonal projections of R upon the 
axes, the Euclidean Hypothesis would hold in the plane. 

8. The functional method applied in 6 to the 
composition of two equal forces acting at a point, is derived 
from D. DE FONCENEX [1734 1799]. By a method ana- 

Rectangular Components of a Fc/ce. IQJ 

logous to that which led us to the equation for / (a) (= y), 
FONCENEX arrived at the differential equation x 

From this, on integrating and taking account of the initial 
conditions of the problem, he obtained the known expression 
for/ (a). 

However the application of the principles of the In 
finitesimal Calculus, requires the continuity and differentiabil 
ity of /(a), conditions, which, as FONCENEX remarks, involve 
the (physical) nature of the problem. But as he wishes to 
go jusqu aux difficultes les moins fondees , he avails himself 
of the Calculus of Finite Differences, and of a Difference 
Equation, which allows him to obtain / (a) for all values of 
a which are commensurable with IT. The case a incom 
mensurable is treated par une methode familiere aux Geo- 
metres et frequente surtout le ecrits des Anciens ; that is, by 
the Method of Exhaustion. 2 

All FONCENEX argument, and therefore that given in 

1 We could obtain this equation from (l) p. 189 as follows: 
Put p==^/a and suppose that /"(a) can be expanded by TAYLOR S 
Series for every value of a. 

Then we have 

2/(a) ^(0) + i/a/ (o) + ^ f" (o) . . .) 

- 2/(o) + 2 - 02 /" (O) + - 

Equating the coefficients of da* and putting y = /(a) and /* 
= f" (o), we have 

2 Cf. FONCENEX : Sur les principes Jondamentattx de la Mccan- 
ique. Misc. Taurinensia. T. II, p. 305315 (17601761). His 
argument is repeated and explained by A. GENOCCHI in his paper: 
Sur uti Memoirs de Daz iet de Foncenex et sur les geometries non- 
euclidiennes. Torino, Memorie (2), T. XXIX, p. 366 371 (1877). 

Appendix I. The Fundamental Principles of Statics etc. 

6, is independent of EUCLID S Postulate. However, it 
should be remarked that FONCENEX aim was not to make 
the law of composition of concurrent forces independent of 
the theory of parallels, but rather to prove the law itself. 
Probably he held, as other geometers [D. BERNOUILLI, 
D ALEMBERT], that it was a truth independent of any ex 
perimental foundation. 

Non-Euclidean Statics. 

9. Having thus shown that the analytical law for 
the composition of concurrent forces does not depend on 
EUCLID S Fifth Postulate, we proceed to deduce the law accord 
ing to which forces perpendicular to a line will be composed. 

Let A, A be the points of application of two torces 
P lt P 2 of equal magnitude P (cf. Fig. 69). 




Fig. 69. 

Let C be the middle point of AA, and B a point on 
the perpendicular BC to AA . 

Joining AB and A B, and putting 

< BAG - 0, < ABC = p, 

it is clear that the force /\ can be regarded as a component 
of a force T lf acting at A and along BA. 

The magnitude of this force is given by 

T= * 

sin a 

Equal Forces perpendicular to a Line. IQ^ 

The other component Q lt at right angles to P lt is 
given by 

Q = Tcos a = P cot a. 

Repeating this process with the force P 2 , we obtain the 
following system of coplanar forces: 

(1) System />,, P*. 

(2) System />,P 2 , &, ft- 

(3) System T lt T 2 . 

If we assume that we can move the point of application 
of a force along its line of action, it is clear that the first two 
systems are equivalent, and because (2) is equivalent to (3), 
we can substitute for the two forces P lt P 2 , the two forces 
71 and T 2 . 

The latter, being moved along their lines of action to B, 
can be composed into one force 

This, in its turn, can be moved to C, its direction per 
pendicular to A A remaining unchanged. 

This result, which is obviously independent of EUCLID S 
Postulate, can be applied to the three systems of geometry: 

Euclid s Geometry. 

In the triangle ABC we have 

cos p = sin a. 

R = 2 P. 

Geometry of Lobatschewsky-Bolyai. 
In the triangle ABC, if we denote the side A A by 2 /-, 
we have 

cos 6 b 

= cosh (p. ii7). 


R =- 2 P cosh 

194 Appendix I. The Fundamental Principles of Statics etc. 

Riemann s Geometry. 

In the same triangle we have 

cos B b 

sin a k 


R = 2 P cos y 


It is only in Euclidean space that the resultant of two 
equal forces, perpendicular to the same line, is equal to the 
sum of the two given forces. In the Non-Euclidean spaces 
the resultant depends, in the manner indicated above, on 
the distance between the points at which the two forces are 
applied. 1 

10. The case of two unequal forces P, Q, per 
pendicular to the same straight line, is treated in a similar 

In the Euclidean Geometry we obtain the known results; 

R = P + Q, 
R P _ Q 

In the Geometry of LOBATSCHEWSKY-BOLYAI the problem 
of the resultant leads to the following equations: 

R = P cosh + Q cosh -|-, 
R P Q 


Then, by the usual substitution of the circular functions 
for the hyperbolic, we obtain the corresponding result for 
RIEMANN S Geometry: 

1 For a fuller treatment of Non-Euclidean Statics, the reader 
is referred to the following authors: J. M. DE TILLY, Etudes de 
Mecanique abstraite, Mem. couronnes et autres m6m., T. XXI (1870). 
J. ANDRADE, La Statique et les Geometries de Lobatsch&vsky , d Euclide, 
et dc Riemann. Appendix (II) of the work quoted on p. 181. 

Unequal Forces. 
P COS y 4- (2 COS - 


/ + <J -9 P 

sin -|- sin -j sin -j 

In these formulae /, ^, denote the distances of the 
points of application of jPand Q from that of It. 

These results can be summed up in a single formula, 
valid for Absolute Geometry, 

R = P. Ep + Q. Eq, 
R P Q 

To obtain these results directly, it is sufficient to use the 
formulae of Absolute Trigonometry, instead of the Euclidean 
or Non-Euclidean, in the argument of which a sketch has 
just been given. 

Deduction of Plane Trigonometry from Statics. 

ii. Let us see, in conclusion, how it is possible to 
treat the converse question : given the law of composition of 
forces, to deduce the fundamental equations of trigonometry. 

To this end we note that the magnitude of the resultant 
R of two equal forces P, perpendicular to a line A A of 
length 2 fr, will in general be a function of P and b. 

Denoting this function by 

9 (P, *), 

we have 

* = <P (P, *), 
or more simply 1 

i The proportionality of R and P follows from the law of 
association on which the composition of forces depends. In fact, 
let us imagine each of the forces /*, acting at A and A , to be 


Io6 Appendix I. The Fundamental Principles of Statics etc. 

On the other hand in 9 (p. 193), we were brought to 
the following expression for R: 

sin a 
Eliminating R and P^ between these, we have 

/z\ cos 6 

m (b\ = - H . 
sm a 

Thus if the analytical expression for q> (b) is known, 
this formula will supply a relation between the sides and 
angles of a right-angled triangle. 

To determine q> (), it is necessary to establish the 
corresponding functional equation. 

With this view, let us apply perpendicularly to the line 
AA, the four equal forces P I} P 2 ,P^,P^, in such a way that 
the points of application of P^ and /* 4 , P 2 and P^ are 
distant 2 (a + ff) and 2 ( #), respectively (cf. Fig. 70). 

We can determine the resultant R of these four forces 
in two different ways: 

(i) Taking P^ with P 2) and P 3 with P^, we obtain two 
forces R vy R 2 of magnitude: 

replaced by equal forces, applied at ^4 and A . Combining 
these, we would have for R the expression 

Comparing this result with the equation given in the text, we have 

v( P * ^} = <P(^*)- 
\n J n 

Similarly we have 

q> (kP. b] - ^cp (/> ^), 

for every rational value of ; and the formula may be extended 
to irrational values. 

Then putting P = I and k = P we obtain 

Q. E. D. 

Deduction of Trigonometry from Statics. 


and taking JR lt R 2 together, we obtain 

R = Py (a) qp (b). 
(ii) Taking /* x with P^ we obtain a force of magnitude : 

P <P (b + ), 
and taking P 2 with ^3, we obtain another of magnitude: 

Taking these two together we have, finally, 
a) + P qp (b a). 












Fig. 70. 

From the two expressions for R we obtain the functional 
equation which (p (b) satisfies, namely, 

(2) <p() 9() = 9(^ + ) + cp(^ a). 

This equation, if we put op (b) = 2/(b), is identical 
with that met in 6 (p. 189), in treating the composition of 
concurrent forces. 

The method followed in finding (2) is due to D ALEM- 
BERT. 1 However, if we suppose a and b equal to each other, 
and if we note that <p (<?) = 2, the equation reduces to 

(3) [<?(*)? = <P(2*)+ 2. 

This last equation was obtained previously by FONCENEX, 
in connection with the equilibrium of the lever. 2 

x Opuscules mathcmatiques, T. VI, p. 371 (1779). 

2 Cf. p. 319322 of the work by FONCENEX, referred to 


IQ8 Appendix I. The Fundamental Principles of Statics etc. 

12. The statical problem of the composition of 
forces is thus reduced to the integration of a functional 

FONCENEX, who was the first to treat it in this way 1 , 
thought that the only solution of (3), was cp (x) = const. If 
this were so, the constant would be 2, as is easily verified. 

Later LAPLACE and D ALEMBERT integrated (3), obtaining 

where c is a constant, or any function which takes the same 
value when x is changed to 2 x. 2 

The solution of LAPLACE and D ALEMBERT, applied to 
the statical problem of the preceding section, leads to the 
case in which c is a function of x. Further, since we cannot 
admit values of c such as a + i b, where a, b are both different 
from zero, we have three possible cases, according as c is 
real, a pure imaginary, or infinite.^ Corresponding to these 

1 We have stated above (p. 53), when speaking of FONCENEX 
memoir, that, if it was not the work of Lagrange, it was certainly 
inspired by him. This opinion, accepted by GENOCCHI and other 
geometers, dates from DELAMBRE. The distinguished biographer 
of LAGRANGE puts the matter in the following words: "// (La- 
grange) fournissait a Foncenex la partie analytique de ses memoir es en 
liei laissant le soin de developper les raisonnements sur lesquels portaient 
ses formules. En ejfet, on remarque dya dans ces me moires (of 
FONCENEX) cette mar che purement analitique, qui depuis a fait le 
caractere des grandes productions de Lagrange. II avail trouve tine 
nouvellc theorie du levier". Notices sur la voie et les outrages de M. 
le Comte Lagrange. Mem. Inst. de France, classe Math, et Physique, 
T. XIII, p. XXXV (1812). 

2 Cf. D ALEMBERT: Sur les principes de la Mecanique : Me"m. de 
1 Ac. des Sciences de Paris (1769). LAPLACE: Recherches sur 
rintegration des equations dtffirentielles : Mem. Ac. sciences de Paris 
(savants etrangers) T. VII (1733). Oeuvres de Laplace, T. VIII, 
p. 1067. 

3 We can obtain this result directly by integrating the equa- 

The Three Geometries. 

I 99 

three cases, we have three possible laws for the composition 
of forces, and consequently three distinct types of equations 
connecting the sides and angles of a triangle. These results 
are brought together in the following table, where k denotes 
a real positive number. 

Value of c 

Form of qp (x) 

cal equations 

Nature of 


X X 


cos p 
cosh T"sin a 


c = ik 

i x ix 
e k-\-e = 2 cos 

cos p 
COS T = sina 


C 00 

X X 

- fto i eo 

i= cosj 


sin a 

Conclusion: The law for the composition offerees per 
pendicular to a straight line, leads, in a certain sense, to the 
relations which hold between the sides and angles of a 
triangle, and thus to the geometrical properties of the plane 
and of space. 

This fact was completely established by A. GENOCCHI 
[1817 1889] in two most important papers 1 , to which the 
reader is referred for full historical and bibliographical 
notes upon this question. 

tion (2), or, what amounts to the same thing, equation (l) of 
S 6. Cf., for this, the elementary method employed by CAUCHY 
for finding the function satisfying (i). Oeuvres de Cauchy, (ser. 2). 
T. Ill, p. 106-113. 

1 One of them is the Memoir referred to on p. 191. The 
other, which dates from 1869, is entitled: Dei primi prindpii delta 
meccanica e della geometria in relazione al postulate d Euclide. Annali 
della Societa italiana delle Scienze (3). T. II, p. 153189. 

Appendix II. 

Clifford s Parallels and Surface. 
Sketch of Clifford-Klein s Problem. 

Clifford s Parallels. 

i. EUCLID S Parallels are straight lines possessing the 
following properties: 

a) They are coplanar. 

b) They have no common points. 

c) They are equidistant. 

If we give up the condition (c) and adopt the views of 
GAUSS, LOBATSCHEWSKY and BOLYAI, we obtain a first ex 
tension of the notion of parallelism. But the parallels which 
correspond to it have very few properties in common with 
the ordinary parallels. This is due to the fact that the most 
beautiful properties we meet in studying the latter depend 
principally on the condition (c). For this reason we are led 
to seek such an extension of the notion of parallelism, that, 
so far as possible, the new parallels shall still possess the 
characteristics, which, in Euclidean geometry, depend on 
their equidistance. Thus, following W. K. CLIFFORD [1845 
1879], we g* ve U P tne property of coplanarity, in the definition 
of parallels, and retain the other two. The new definition of 
parallels will be as follows: 

Two straight lines, in the same or in different planes , are 
called parallel, when the points of the one are equidista?it from 
the points of the other. 

Clifford s Parallels. 2OI 

2. Two cases, then, present themselves, according as 
these parallels lie, or^do not lie, in the same plane. 

The case in which the equidistant straight lines are 
coplanar is quickly exhausted, since the discussion in the 
earlier part of this book [ 8] allows us to state that the 
corresponding space is the ordinary Euclidean. We shall, 
therefore, suppose that the two ^ ^ 

equidistant straight lines r and s T 
are not in the same plane, and 
that the perpendiculars drawn 
from r to s are equal. Obvi- 5 - 

ously these lines will also be per- 

Fig. 71. 

pendicular to r. Let AA , BB 

be two such perpendiculars (Fig. 71). The skew quad 
rilateral ABB A, which is thus obtained, has its four angles 
and two opposite sides equal. It is easy to see that the 
other two opposite sides AB, A B are equal, and that the 
interior alternate angles, which each diagonal e. g. AB 
makes with the two parallels, are equal. This follows from 
the congruence of the two right-angled triangles AAB and 
ABB . 

If now we examine the solid angle at^, from a theorem 
valid in all the three geometrical systems, we can write 

< AAB -f < B AB > < AAB = i right angle. 

This inequality, taken along with the fact that the angles 
AB A and B AB are equal, can be written thus: 

<; AAB + ^ AB A > i right angle. 

Stated in this way, we see that the sum of the acute 
angles in the right-angled triangle AA B is greater than a 
right angle. Thus in the said triangle the Hypothesis of the 
Obtuse Angle is verified, and consequently parallels not in the 
same plane can exist only in the space of Riemann. 


Appendix II. Clifford s Parallels and Surface. 

3. Now to prove that in the elliptic space of RIEMANN 
there actually do exist pairs of straight lines, not in the same 
plane and equidistant, let us consider an arbitrary straight 
line r and the infinite number of planes perpendicular to it. 
These planes all pass through another line r, the polar 
of r in the absolute polarity of the elliptic space. Any line 
whatever, joining a point of r with a point of/, is perpend 
icular both to r and to r , and has a constant length, equal 
to half the length of a straight line. From this it follows 
that r, r are two equidistant straight lines, not in the same 

But two such equidistants represent a very particular 
case, since all the points of r have the same distance not 
only from r , but from all the points ofr. 



K B| 

Fig. 72. 

To establish the existence of straight lines in which the 
last peculiarity does not exisfe, we consider again two lines 
r and r , one of which is the polar of the other (Fig. 72). 
Upon these let the equal segments AB, AB be taken, each 
less than half the length of a straight line. Joining A with 
A , and B with B , we obtain two straight lines a, b, not 
polar the one to the other, and both perpendicular to the 
lines r, r . 

It can easily be proved that a t b are equidistant. To 
show this, take a segment AH upon AA } then on the 

The Polars as Parallels. 


supplementary line r to AHA, take the segment AM equal to 
AH. If the poinfs H and M are joined respectively with 
B and B, we obtain two right-angled triangles AB H, ABM, 
which, in consequence of our construction, are congruent. 

We thus have the equality 

HB = BM. 

Now if H and B are joined, and the two triangles 
HBB and HBMatQ compared, we see immediately that they 
are equal. They have the side HB common, the sides HB 
and MB equal, by the preceding result, and finally BB and 
HM are also equal, each being half of a straight line. 

This means, in other words, that the various points of 
the straight line a are equidistant from the line b. Now since 
the argument can be repeated, starting from the line b and 
dropping the perpendiculars to a, we conclude that the line 
HK, in addition to being perpendicular to , is also perpend 
icular to a. 

We remark, further, that from the equality of the 
various segments AB, HK, A B , ... the equality of the re 
spective supplementary segments is deduced, so that the two 
lines a, b, can be regarded as equidistant the one from the 
other, in two different ways. If then it happened that the 
line AB were equal to its supplement, we would have the ex 
ceptional case, which we noted previously, where a, b are 
the polars of each other, and consequently all the points of 
a are equidistant from the different points of b. 

4. The non-planar parallels of elliptic space were 
discovered by CLIFFORD in 1873. 2 Their most remarkable 
properties are as follows: 

1 The two different segments, determined by two points on 
a straight line, are called supplementary. 

2 Preliminary Sketch of Biquaternhns. Proc. Lond. Math. Soc. 
Vol. IV. p. 381 395(1873). Clifford s Mathematical Papers, p. 181 2OO. 


Appendix II. Clifford s Parallels and Surface. 

(i) If a straight line meets two parallels, it makes with 
them equal corresponding angles, equal interior alternate 
angles, etc. 

(ii) If in a skew quadrilateral the opposite sides are 
equal and the adjacent angles supplementary, then the opposite 
sides are parallel. 

Such a quadrilateral can therefore be called a skew 
parallelogram . 

The first of these two theorems can be immediately 
verified; the second can be proved by a similar argument 
to that employed in 3. 

(iii) If two straight lines are equal and parallel, and 
their extremities are suitably joined, we obtain a skew paral 

This result, which can be looked upon, in a certain 

sense, as the converse of (ii), can also be readily established. 

(iv) Through any point (M) in space, which does not 

lie on the polar of a straight line (r}, two parallels can be 

drawn to that line. 

Indeed, let the perpendicular MN be drawn from M 
to ;-, and let N be the point in which the polar of MN 

meets r (Fig. 73). From 
this polar cut off the two 
segments N M , N M" , 
equal to NM, and join the 
points M , M" to M. The 
two lines r , r", thus ob 
tained, are the required par 


Fig- 73- 

If M lay on the polar of r, then MN would be 
equal to half the straight line; the two points M , M" 
would coincide: and the two parallels r , r" would also 

Properties of Clifford s Parallels. 


The angle between the two parallels /, r" can be 
measured by the segment M M", which the two arms of the 
angle intercept on the polar of its vertex. In this way we 
can say that half of the angle between r and / , that is, 
the angle of parallelism, is equal to the distance of parallelism. 

To distinguish the two parallels r , r", let us consider a 
helicoidal movement of space, with MN for axis, in which 
the pencil of planes perpendicular to MN, and the axis MM" 
of that pencil, obviously remain fixed. Such a movement 
can be considered as the resultant of a translation along MN, 
accompanied by a rotation about the same axis: or by two 
translations, one along MN, the other along M M". If the 
two translations are of equal amount, we obtain a space 

.Vectors can be right-handed or left-handed. Thus, referr 
ing to the two parallels r , r" , it is clear that one of them 
will be superposed upon r by a right-handed vector of 
magnitude MN, while the other will be superposed on r by 
a left-handed vector of the same magnitude. Of the two 
lines r, r", one could be called the right-handed parallel 
and the other the left-handed parallel to r. 

(v) Two right-handed (or left-handed) parallels to a 
straight line are right-handed (or left-handed} parallels to 
each other. 

Let b, c be two right-hand 
ed parallels to a. From the 
two points A, A of a, distant 
from each other half the length 
of a straight line, draw the 
perpendiculars AB, AB on b, 
and the perpendiculars AC, 
AC on c (cf. Fig. 74). 

The lines AB , AC are the polars of AB and AC. 

Therefore <C BAC 

2O6 Appendix II. Clifford s Parallels and Surface. 

Further, by the properties of parallels 

Therefore the triangles ABC, A B C are equal 

Thus it follows that 

BC = B C. 

Again, since 

BB = AA = CC, 
the skew quadrilateral BB C C has its opposite sides equal. 

But to establish the parallelism of , c, we must also 
prove that the adjacent angles of the said quadrilateral are 
supplementary (cf. ii). For this we compare the two solid 
angles B (AB C) and B (AB" C }. In these the following 
relations hold: 

^ABB = ^c AB B" = i right angle 
^ ABC = < A JffC . 

Further, the two dihedral angles, which have BA and 
B A for their edges, are each equal to a right angle, dimin 
ished (or increased) by the dihedral angle whose normal 
section is the angle A BB . 

Therefore the said two solid angles are equal. From 
this the equality of the two angles B BC, B" B C follows. 
Hence we can prove that the angles B, B of the quadri 
lateral BB C C are supplementary, and then (on drawing 
the diagonals of the quadrilateral, etc.) that the angle B is 
supplementary to C, and C supplementary to C", etc. 

Thus b and c are parallel. From the figure it is clear 
that the parallelism between b and c is right-handed, if that 
is the nature of the parallelism between the said lines and 
the line a. 

Clifford s Surface. 

5. From the preceding argument it follows that all 
the lines which meet three right-handed parallels are left-handed 
parallels to each other. 

Clifford s Surface. 


Indeed, if ABC is a transversal cutting the three lines 
a, b, c, and if three equal segments A A , BB , CC are taken 
on these lines in the same direction, 1 the points A B C lie 
on a line parallel to ABC. The parallelism between ABC 
and AB C is thus left-handed. 

From this we deduce that three parallels a, b, c, define 
a ruled surface of the second order (CLIFFORD S Surface). 
On this surface the lines cutting a, b, c form one system of 
generators (g^)\ the second system of generators (g<t) is 
formed by the infinite number of lines, which, like a, b, c, 
meet (>,). 

CLIFFORD S Surface possesses the following charact 
eristic properties: 

a) Two generators of the same system are parallel to 
each other. 

b) Two generators of opposite systems cut each other at a 
constant angle. 

6. We proceed to show that Clifford s Surface has 
two distinct axes of revolution. 

To prove this, from 
any point M draw the g 

parallels d (right-hand 
ed), s (left-handed), to a 
line r, and denote by 6 
the distance MN of 
each parallel from r 
(cf. Fig. 75). 

Keeping,/ fixed, let 
s rotate about r, and let /, j", /" ... be the successive 
positions which s takes in this rotation. 

1 It is clear that if a direction is fixed for one line, it is 
then fixed for every line parallel to the first. 

2O8 Appendix II. Clifford s Parallels and Surface. 

It is clear that j, /, s" . . . are all left-handed parallels 
to r and that all intersect the line d. 

Thus s in its rotation about r generates a CLIFFORD S 

Vice versa, if d and s are two generators of a CLIFFORD S 
Surface, which pass through a point M of the surface, and 2 b 
the angle between them, we can raise the perpendicular 
to the plane sd at M and upon it cut off the lines 
ML = MN = 6. 

Let D and S be the points where the polar of LN meets 
the lines d and j, respectively, and let H be the middle point 
of>S*= 20. 

Then the lines HL and HN are parallel, both to s 

Of the two lines HL and HN choose that which is 
a right-handed parallel to d and a left-handed parallel to s t 
say the line HN. 

Then the given CLIFFORD S Surface can be generated by 
the revolution of s or d about HN. 

In this way it is proved that every CLIFFORD S Surface 
possesses one axis of rotation and that every point on the 
surface is equidistant from it. 

The existence of another axis of rotation follows im 
mediately, if we remember that all the points of space, equi 
distant from HN) are also equidistant from the line which is 
the polar of HN. 

This line will, therefore, be the second axis of rotation 
of the CLIFFORD S Surface. 

7. The equidistance of the points of CLIFFORD S 
Surface from each axis of rotation leads to another most 
remarkable property of the surfaces. In fact, every plane 
passing through an axis r intersects it in a line equidistant 
from the axis. The points of this line, being also equally 
distant from the point (O) in which the plane of section meets 

The Axes of Clifford s Surface. 

the other axis of the surface, lie on a circle, whose centre (O) 
is the pole of r with respect to the said line. Therefore the 
meridians and the parallels of the surface are circles. 

The surface can thus be generated by making a circle 
rotate about the polar of its centre, or by making a circle move 
so that its centre describes a straight line, while its plane is 
maintained constantly perpendicular to it (BiANCHi). 1 

This last method of generating the surface, common 
also to the Euclidean cylinder, brings out the analogy be 
tween CLIFFORD S Surface and the ordinary circular cylinder 
This analogy could be carried further, by considering the 
properties of the helicoidal paths of the points of the surface, 
when the space is submitted to a screwing motion about 
either of the axes of the surface. 

8. Finally, we shall show that the geometry on CLIF 
FORD S Surface, understood in the sense explained in SS 67, 
68, is identical with Euclidean geometry. 

To prove this, let us determine the law according to 
which the element of distance between two points on the 
surface is measured. 

Let u, v, be respectively a parallel and a meridian 
through a point O on the surface, and M any arbitrary point 
upon it. 

Let the meridian and parallel 
through M cut off the arcs OP, OQ 
from u and v. The lengths u, v of 
these arcs will be the coordinates of Q 
M. The analogy between the system 
of coordinates here adopted and the 
Cartesian orthogonal system is evident 
(cf. Fig. 76). Fi *- 75. 

1 Sulla superficie a curvatura nulla in geometria ellittica. Ann. 
di Mat. (2) XXIV, p. 107 (1896). Also Lezisni di Geometria Differ- 
enziale. 2a Ed., Vol. I, p. 454 (Pisa, 1902). 


2JO Appendix II. Clifford s Parallels and Surface. 

Let M be a point whose distance from M is infini 
tesimal. If (u, v) are the coordinates of Af t we can take 
(u -t- du, v -\- dv) for those of M . 

Now consider the infinitesimal triangle MM N, whose 
third vertex N is the point in which the parallel through M 
intersects the meridian through M . It is clear that the angle 
MNM is a right angle, and that the sides MN t NM are 
equal to du^ dv. 

On the other hand, this triangle can be regarded as 
rectilinear (as it lies on the tangent plane at M}. So that, 
from the properties of infinitesimal plane triangles, its hypo 
tenuse and its sides, by the Theorem of Pythagoras, are con 
nected by the relation 

ds* = du 2 + dv 2 . 

But this expression for ds* is characteristic of ordinary 
geometry, so that we can immediately deduce that the pro 
perties of the Euclidean plane hold in every normal region on 
a Clifford s Surface. 

An important application of this result leads to the 
evaluation of the area of this surface. Indeed, if we ^break 
it up into such congruent infinitesimal parallelograms by 
means of its generators, the area of one of these will be 
given by the ordinary expression 

dx dy sin 0, 

where dx, dy are the lengths of the sides and 6 is the con 
stant angle between them (the angle between two generators). 
The area of the surface is therefore 

^L dx dy sin 6 = sin 9 2 dx Z dy. 

But both the sums S dx, Z dy represent the length / of 
a straight line. 

Therefore the area A of CLIFFORD S Surface takes the 
very simple form, 

The Area of Clifford s Surface. 211 

A = I 2 sin 0, 

which is identical with the expression for the area of a 
Euclidean parallelogram (CLIFFORD). 1 

Sketch of Clifford-Klein s Problem. 

9. CLIFFORD S ideas, explained in the preceding 
sections, led KLEIN to a new statement of the fundamental 
problem of geometry. 

In giving a short sketch of KLEIN S views, let us refer 
to the results of 68 regarding the possibility of interpret 
ing plane geometry by that on the surfaces of constant 
curvature. The contrast between the properties of the Eu 
clidean and Non-Euclidean planes and those of the said 
surfaces was there restricted to suitably bounded regions. 
In extending the comparison to the unbounded regions , we 
are met, in general, by differences; in some cases due to 
the presence of singular points on the surfaces (e. g., vertex 
of a cone) ; in others, to the different connectivities of the 

Leaving aside the singular points, let us take the cir 
cular cylinder as an example of a surface of constant curv 
ature, every where regular, but possessed of a connectivity 
different from that of the Euclidean plane. 

The difference between the geometry of the plane and 
that of the cylinder, both understood in the complete sense, 
has been already noticed on p. 140, where it was observed 
that the postulate of congruence between two arbitrary 
straight lines ceases to be true on the cylinder. Nevertheless 
there are numerous properties common to the two geometries, 

1 Preliminary Sketch, cf. p. 203 above. The properties of 
this surface were referred to only very briefly by CLIFFORD in 1873. 
They are developed more fully by KLEIN in his memoir: Zui- nicht- 
cuklidischen Gcometrie, Math. Ann. Bd. XXXVII, p. 544572 (1890). 


212 Appendix II. Clifford s Parallels and Surface. 

which have their origin in the double characteristic, that 
both the plane and the cylinder have the same curvature, 
and that they are both regular. 

These properties can be summarized thus: 

1) The geometry of any normal region of the cylinder 
is identical with that of any normal region of the plane. 

2) The geometry of any normal region whatsoever of 
the cylinder, fixed with respect to an arbitrary point upon it, 
is identical with the geometry of any normal region what 
soever of the plane. 

The importance of the comparison between the ge 
ometry of the plane and that of a surface, founded on the 
properties (i) and (2), arises from the following consid 
erations : 

A geometry of the plane, based upon experimental 
criteria, depends on two distinct groups of hypotheses. The 
first group expresses the validity of certain facts, directly 
observed in a region accessible to experiment {postulates of 
the normal region) ; the second group extends to inaccessible 
regions some properties of the initial region (postulates of 

The postulates of extension could demand, e. g., that 
the properties of the accessible region should be valid in the 
entire plane. We would then be brought to the two forms, 
the parabolic and the hyperbolic plane. If, on the other hand, 
the said postulates demanded the extension of these pro 
perties, with the exception of that which attributes to the 
straight line the character of an open line, we ought to take 
account of the elliptic plane as well as the two planes mentioned. 

But the preceding discussion on the regular surfaces of 
constant curvature suggests a more general method of enun 
ciating the postulates of extension. We might, indeed, simply 
demand that the properties of the initial region should hold 
in the neighbourhood of every point of the plane. In this 

Clifford-Klein s Problem. 


case, the class of possible forms of planes receives con 
siderable additions. We could, e. g., conceive a form with 
zero curvature, of double connectivity, and able to be com 
pletely represented on the cylinder of Euclidean space. 

The object of Cliff or d- Klein s problem is the determination 
of all the two dimensional manifolds of constant curvature, 
which are everywhere regular. 

10. Is it possible to realise, with suitable regular 
surfaces of constant curvature, in the Euclidean space, all 
the form s of CLIFFORD-KLEIN ? 

The answer is in the negative, as the following example 
clearly shows. The only regular developable surface of the 
Euclidean space, whose geometry is not identical with that 
of the plane, is the cylinder with closed cross-section. On 
the other hand, CLIFFORD S Surface in the elliptic space is a 
regular surface of zero curvature, which is essentially different 
from the plane and cylinder. 

However with suitable conventions we can represent 
CLIFFORD S Surface even in ordinary space. 

Let us return again to the cylinder. If we wish to un 
fold the cylinder, we must first render it simply connected 
by a cut along a generator (g) , then, by bending without 
stretching, it can be spread out on the plane, covering a 
strip between two parallels (gi,g 2 ). 

There is a one-one correspondence between the points 
of the cylinder and those of the strip. The only exception is 
afforded by the points of the generator (g\ to each of which 
correspond two points, situated the one on^, the other on 
g 2 . However, if it is agreed to regard these two points as 
identical, that is, as a single point, then the correspondence 
becomes one-one without exception, and the geometry of the 
strip is completely identical with that of the cylinder. 

21 A Appendix II. Clifford s Parallels and Surface. 

A representation analogous to the above can also be 
adopted for CLIFFORD S Surface. First the surface is made 
simply connected by two cuts along the intersecting gener 
ators (g, g ). In this way a skew parallelogram is obtained 
in the elliptic space. Its sides have each the length of a 
straight line, and its angles 6 and 6 [0 + 6 = 2 right angles] 
are the angles between g and g. 

This being done, we take a rhombus in the Eu 
clidean plane, whose sides are the length of the straight line 
in the elliptic plane, and whose angles are 0, . On this 
rhombus CLIFFORD S Surface can be represented congruently 
(developed). The correspondence between the points of the 
surface and those of the rhombus is a one-one correspond 
ence, with the exception of the points Q{ g and^- , to each 
of which correspond two points, situated on the opposite 
sides of the rhombus. However, if we agree to regard these 
points as identical, two by two, then the correspondence 
becomes one-one without exception, and the geometry of 
the rhombus is completely identical with that of Clifford s 

ii. These representations of the cylinder and of 
CLIFFORD S Surface show us how, for the case of zero curva 
ture, the investigation of CLIFFORD-KLEIN S forms can be 
reduced to the determination of suitable Euclidean polygons, 
eventually degenerating into strips, whose sides are two by 
two transformable, one into the other, by suitable movements 
of the plane, their angles being together equal to four right- 
angles (KLEIN). 2 Then it is only necessary to regard the 
points of these sides as identical, two by two, to have a 
representation of the required forms on the ordinary plane. 

1 Cf. CLIFFORD loc. cit. Also KLEIN S memoir referred to 
on p. 211. 

> Cf. the memoir just named. 

Clifford-Klein s Problem. 21 5 

It is possible to present, in a similar way, the investi 
gation of CLIFFORD-KLEIN S forms for positive or negative 
values of the curvature, and the extension of this problem 
to space. 1 

i A systematic treatment of CLIFFORD-KLEIN S problem is to 
be found in KILLING S Einfuhrung in die Grundlagen der Geometric. 
Bd. I, p. 271 349 (Paderborn, 1893). 

Appendix III. 

The Non-Euclidean Parallel Construction 
and other Allied Constructions. 

i. The Non-Euclidean Parallel Construction depends 
upon the correspondence between the right-angled triangle 
and the quadrilateral with three right angles. Indeed, when 
this correspondence is known, a number of different con 
structions are immediately at our disposal. f 

To express this correspondence we introduce the 
following notation: 

In the right-angled triangle, as usual, #, b are the sides: 
c is the hypotenuse: X is the angle opposite a and JLX 
that opposite b. Further the angles of parallelism for a, b 
are denoted by a and P : and the lines which have X, ju for 
angles of parallelism are denoted by /, m. Also two lines, 
for which the corresponding angles of parallelism are com 
plementary, are distinguished by accents, e. g.: 

TTGO-~TT( a ), TT(/)-y--TTM. 

Then with this notation: To every right-angled triangle 
(a, b, fj X, ju) there corresponds a quadrilateral with three 
right-angles, whose fourth angle (acute) is p, and whose sides 
are c, m , a, /, taken in order frotn the corner at which the 
angle is p. 

The converse of this theorem is also true. 

1 Cf. p. 256 of ENGEL S work referred to on p. 84. 

Correspondence between Quadrilateral and Triangle. 217 

The following is one of the constructions, which can be 
derived from this theorem, for drawing the parallel through 
A to the line BC (cf. Fig. 77). 

Let AB be the perpendicular from A to BC. At A draw 
the line perpendicular to AB, and from any point C in BC 
draw the perpendicular CD 
to this line. 

With centre A and rad 
ius BC (equal to c) describe 
a circle cutting CD in E. 

Now we have 

Fig. 77. 

< EAD = M, 
and therefore 

<: BAE = - |U = FT (;//). 

But the sides of the quadrilateral are c, m , a, /, taken in 
order from C. 

Therefore AE is parallel to BC. 

If a proof of this construction is required without using 
the trigonometrical forms, one might attempt to show direct 
ly that the line AE produced, (simply owing to the equality 
of BC and AE), does not cut BC produced, and that the 
two have not a common perpendicular. If this were the 
case, they would be parallel. Such a proof has not yet been 

Again, we might prove the truth of the construction 
using the theorem, that in a prism of triangular section the 
sum of the three dihedral angles is equal to two right angles 1 : 
so that for a prism with n angles the sum is (2 n 4) right 
angles. This proof is given in 2 below. 

1 Cf. LOBATSCHEWSKY (ENGEL s translation) p. 172. 

2l8 Appendix III. The Non-Euclidean Parallel Construction. 

Finally, the correspondence stated in the above theorem 
only part of which is required for the Parallel Construction 
of Fig. 78 can be verified without the use of the geo 
metry of the Non-Euclidean space. This proof is given in 3. 

2. Direct proof of the Parallel Construction by means 
of a Prism. 


Let ABCD be a plane quadrilateral in which the angles 
at D, A t B are right angles. Let the angle at C be denoted 
by P, AD by a, Z>Cby /, CB by c, and BA by ;// . 

At A draw the perpendicular AQ to the plane of the 
quadrilateral. Through B, C, and D draw BQ, CQ and Z>Q 
parallel to AQ. 

Also through A draw AQ parallel to BC, cutting CD 
in E (ED = 0, and let the plane through JQ and AE 
cut CZ)Q in Q. From the definition, we have 


Further the plane QAB is at right angles to a, and the 
plane Q.DA at right angles to /, since QA and ^# are per 
pendicular to a, while QZ> and a are perpendicular to /. 

Direct Proof of the Parallel Construction. 219 

Also <^ ABO. = <: QAB = u - u. 


In the prism Q (ABCD) the faces which meet in QA, 

QD are perpendicular. Also the four dihedral angles 
make up four right angles. It follows that the faces of the 
prism C (DBQ), which meet along CQ, are perpendicular. 
Also it is clear that in E (DQA) the faces which meet in EA 
are perpendicular, while the dihedral angle for the edge CD 
is the same as for ED (thus equal to a). 

We shall now prove the equality of the other dihedral 
angles in these prisms C (DBQ.) and E (DQA) those con 
tained by the faces which meet in CB and AE. 

In the first prism this angle is equal to the angle be 
tween the planes ABCD and CBQ. It is thus equal to 


~ 2 JLI, i. e. it is equal to <C ABQ. 

In the second prism, the angle between the planes 
meeting in EQ belongs also to the prism Q (ADE). In this 
the angle at QZ> is a right-angle, and that at QA is equal 
to jn. Thus the third angle is equal to ^ |u. 

Therefore the prisms C (DBQ) and E (DQA) are 

Therefore <^ CQ = <^ QA, 

and the lines which have these angles of parallelism are 
also equal. 

Thus c = BC and c l = AE 

are equal, which was to be proved. 

Further it follows that 

<^ DEA = <^ >CQ; 
i. e.the angle X,, opposite the side a of the triangle, is given by 

\ t = TT (/; = X. 

Finally <C DCB = ^C DEO. ; 

i. e. 6 = TT (br). or b^ b. 

22O Appendix III. The Non-Euclidean Parallel Construction. 

Thus the correspondence between the triangle and the 
quadrilateral is proved. 1 

3. Proof of the Correspondence by Plane Geometry. 

In the right-angled triangle ABC produce the hypo 
tenuse AB to D, where the perpendicular at D is parallel to 
CB (cf. Fig. 79). 

Fig. 79- 

Then with the above notation 
BD = m. 
Draw through A the parallel to Z>0 and CBQ. 


<C CAB = = TT (), 

and it is also equal to 

X + ^C DA = X + TT (c + m). 
We thus obtain the first of the six following equations. 2 
The third and fifth can be obtained in the same way. The 
second, fourth, and sixth, come each from the preceding, if 
we interchange the two sides a and fr, and, correspondingly 
the angles X and ju. 

1 BONOLA: 1st. Lombardo, Rend. (2). T. XXXVII, p. 255 
258 (1904). The theorem had already been proved by pure 
geometrical methods by F. ENGEL: Bull, de la Soc. Phys. Math. 
de Kasan (2). T. VI (1896); and Bericht d. Kon. Sachs. Ges. d. 
Wiss., Math.-Phys. Klasse, Bd. L, p. 181187 (Leipzig, 1898). 

2 Cf. LOBATSCHEWSKY (Engel s translation), p. 15 16, and 
LIEBMANN, Math. Ann. Bd. LXI, p. 185, (1905). 

Second Proof of the Parallel Construction. 221 

The table for this case is as follows: 
X + TT (c + m) = P, ju + n (c + 1) = a: 
X + p = TTO m\ ji + a = TTfc /); 

TT (+/; + no fl> = yTr, no + *) + n(/ ^)=yTT. 

Similar equations can also be obtained for the quad 
rilateral with three right angles. Some of the sides have to 
be produced, and the perpendiculars drawn, which are 
parallel to certain other sides, etc. 

If we denote the acute angle of the quadrilateral by p,, 
and the sides, counting from it, by c st m^, a lf and / x , we ob 
tain the following table : 

AI + H (c. 4- ; x ) = p xi Tx + H (/, + O = P X ; 

Xi + Pi = n (c, /;/,), TI + Pi = H (/ x x ); 

The second, fourth, and sixth formulae come from inter 
changing d and mi, with / r and ^Zj , as in the right-angled 

Let us now imagine a right-angled triangle constructed 
with the hypotenuse c and the adjacent angle |U: and let the 
remaining elements be denoted by a, t>, \ as above. 

In the same way, let a quadrilateral with three right- 
angles be constructed, in which c is next the acute angle, ;// 
follows c, the remaining elements being a t , / J5 and PL 

Then a comparison of the first and third formulae for 
the triangle, with the first and third for the quadrilateral, 
shows that 

Pi = P, Xi = X. 

The fifth formula of both tables then gives 

di = a. 
Hence the theorem is proved. 

222 Appendix III. The Non-Euclidean Parallel Construction. 

From the two tables it also follows that to a right- 
angled triangle with the elements 

a, b, c, A, H, 
there corresponds a second triangle with the elements 

a, = a , b 1 <= / , ^ = m, X x = -^ (3, fi x = T , 

a result which is of considerable importance in further con 
structions. But we shall not enter into fuller details. 

The possibility of the Non-Euclidean Parallel Construc 
tion, with the aid of the ruler and compass, allows us to 
draw, with the same instruments, the common perpendicular 
to two lines which are not parallel and do not meet each 
other (the non-intersecting lines}, the common parallel to the 
two lines which bound an angle; and the line which is per 
pendicular to one of the bounding lines of an acute angle 
and parallel to the other. We shall now describe, in a few 
words, how these constructions can be carried out, following 
the lines laid down by HILBERT. X 

4. Construction of the common perpendicular to two 
non-intersecting straight lines. 

Fig. 80. 

Let a = AiAi b = *, be two non-intersecting lines; 
that is, lines which do not meet each other, and are not 
parallel (cf. Fig. 80). 

1 Neue Begrundung der Bolyai- Lobatschefskyschen Geometrie. 
Math. Ann. Bd. 57, p. 137 150 (1903). HILBERT S Gmndlagen der 
Geometrie, 2. Aufl., p. 107 et seq. 

Some Allied Constructions. 223 

Let A 1 B^ AB be the perpendiculars drawn from the 
points A* , A upon a to the line , constructed as in ordinary 

If the segments AT.BI , AB, are equal, the perpendicular 
to b from the middle point of the segment B^B is also per 
pendicular to a; so that, in this case, the construction of 
the common perpendicular is already effected. 

If, on the other hand, the two segments A^B^ AB are 
unequal, let us suppose, e. g., that A^ is greater than AB. 

Then cut off from A^B^ the segment A B* equal to AB; 
and through the point A , in the part of the plane in which 
the segment AB lies, let the ray AM be drawn, such that 
the angle B^AM is equal to the angle which the line a 
makes with AB (cf. Fig. 80). 

The ray A M must cut the line a in a point M (cf. 
HILBERT, loc. cit.). From M drop the perpendicular M P 
to b, and from the line a, in the direction A^A, cut off the 
segment A M equal to AM . 

If the perpendicular MP is now drawn to b, we have a 
quadrilateral ABPM which is congruent with the quad 
rilateral A Bif M . 

It follows that MP is equal to M P . 

It remains only to draw the perpendicular to b from 
the middle point of P P to obtain the common perpendicular 
to the two lines a and b. 

5. Construction of the common parallel to two straight 
lines which bound any angle. 

Let a = AO, and b== BO> be the two lines which con 
tain the angle AOB (cf. Fig. Si). From a and b cut off the 
equal segments OA and OB; and draw through A the ray 
b parallel to the line , and through B the ray a parallel to 
the line a. 

224 Appendix III. The Non-Euclidean Parallel Construction. 

Let tf i and bi be the bisectors of the angles contained 
by the lines ab , and ab. 

The two lines A are non-intersecting lines, and their 
common perpendicular ^4^, the construction for which was 
given in the preceding paragraph, is the common parallel to 
the lines which bound the angle AOB. 

\ B > 

Reference should be made to HILBERT S memoir, quot 
ed above, for the proof of this construction. 

6. Construction of the straight line which is perpendi 
cular to one of the lines bounding an acute angle and parallel 
to the other. 

Let a = AO and b = BO, be 
the two lines which contain the acute 
angle AOB; and let the ray b = B O 
be drawn, the image of the line b in 
9 a (cf. Fig. 82). 

Then, using the preceding con 
struction, let the line B^B^ be drawn 
parallel to the two lines which con 
tain the angle BOB. 

This line, from the symmetry of 
the figure with respect to a, is perpendicular to OA. 

It follows that ^x^ i is parallel to one of the lines which 
contain the angle AOB and perpendicular to the other. 

7. The constructions given above depend upon 
metrical considerations. However it is also possible to make 
use of the fact that to the metrical definitions of perpend- 


Fig. 82. 

Projective Constructions. 225 

icularity and parallelism a project! ve meaning can be given 
(S 79); and that projective geometry is independent of the 
parallel postulate ( 80). 

Working on these lines, what will be the construction 
for the parallels through a point A to a given line? 

Let the points P lt P 2 , P^ and /Y, P 2t /V be given 
on so that the points / , /V, /y, are all on the same 
side of P s , P 2 , /3, and 

Join ^4/x, -4jP 2 , AP^ and denote these lines by s lt s 2 , 
and j,. Similarly let AP*, AP 2 t AP^ be denoted by .$-, , 
J 2 and J 3 . Then the three pairs of rays through A, determ 
ine a projective transformation of the pencil (s) into itself, 
the double elements of which are obviously the two parallels 
which we require. These double elements can be constructed 
by the methods of projective geometry. 1 

The absolute is then determined by five points: i. e., by 
five pairs of parallels; and so all further problems of metrical 
geometry are reduced to those of projective geometry. 

If we represent (cf. 84) the LOBATSCHEWSKY-BOLYAI 
Geometry (e. g., for the Euclidean plane) so that the image 
of the absolute is a given conic (not reaching infinity), then 
it has been shown by GROSSMANN 2 that most of the problems 
for the Non-Euclidean plane can be very beautifully and 
easily solved by this translation . However we must not 
forget that this simplicity disappears, if we would pass from 
the translation back to the original text . 

1 Cf. for example, ENRIQUES, Geometria proicttiva, (referred to 
on p. 156) 8 73- 

2 GROSSMANN, Die fundamental** K.mstruktionen der nicht- 
euklidischen Geometrie, Programm der Thurgauischen Kantonschule, 
(Frauenfeld, 1904). 


226 Appendix III. The Non-Euclidean Parallel Construction. 

In the Non-Euclidean plane the absolute is inaccessible, 
and its points are only given by the intersection of pencils 
of parallels. The points outside of the absolute, while they 
are accessible in the translation , cannot be reached in the 
text itself. In this case they are pencils of straight lines, 
which do not meet in a point, but go through the (ideal) 
pole of a certain line with respect to the absolute. 

If, then, we would actually carry out the constructions, 
difficulties will often arise, such as those we meet in the 
translation of a foreign language, when we must often sub 
stitute for a single adjective a phrase of some length. 

Appendix IV. 

The Independence of Projective Geometry 
from Euclid s Postulate. 

i. Statement of the Problem. In the following pages 
we shall examine more carefully a question to which only 
passing reference was made in the text (cf. 80), namely, the 
validity of Projective Geometry in Non-Euclidean Space, since 
this question is closely related to the demonstration of the 
independence of that geometry from the Fifth Postulate. 

In elliptic space (cf. 80) we may assume that the 
usual projective properties of figures are true, since the 
postulates of projective geometry are fully verified. Indeed 
the absence of parallels, or, what amounts to the same thing, 
the fact that two coplanar lines always intersect, makes the 
foundation of projectivity in elliptic space simpler than in Eu 
clidean space, which, as is well known, must be first com 
pleted by the points at infinity. 

However in hyperbolic space the matter is more com 
plicated. Here it is not sufficient to account for the absence 
of the point common to two parallel lines, an exception 
which destroys the validity of the projective postulate: two 
coplanar lines have a common point. We must also remove 
the other exception the existence of coplanar lines which 
do not cut each other, and are not parallel (the non-inter 
secting lines]. The method, which we shall employ, is the 
same as that used in dealing with the Euclidean case. We 
introduce fictitious points, regarded as belonging to two co 
planar lines which do not meet. 


228 App. IV. The Indcpend. of Proj. Geo. from Euclid s Post. 

In the following paragraphs, keeping for simplicity to 
two dimensions only, we show how these fictitious points 
can be introduced on the hyperbolic plane, and how they 
enable us to establish the postulates of protective geometry 
without exception. Naturally no distinction is now made be 
tween the proper points, that is, the ordinary points, and the 
fictitious points, thus introduced. 

2. Improper Points and the Complete Protective Plane. 
We start with the pencil of lines, that is, the aggregate of 
the lines of a plane passing through a point. We note that 
through any point of the plane, which is not the vertex of 
the pencil, there passes one, and only o?ie, line of the pencil. 

On the hyperbolic plane, in addition to the pencil, there 
exist two other systems of lines which enjoy this property, 

(i) the set of parallels to a line in one direction \ 

(ii) the set of perpendiculars to a line. 

If we extend the meaning of the term, pencil of lines, 
we shall be able to include under it the two systems of lines 
above mentioned. In that case it is clear that two arbi 
trary lines of a plane will determine a pencil, to which they 

If the two lines are concurrent, the pencil is formed by 
the set of lines passing through their common point; if they 
are parallel, by the set of parallels to both, in the same 
direction; finally, if they are non-intersecting, by all the lines 
which are orthogonal to their common perpendicular. In 
the first type of pencil (the proper pencil], there exists a point 
common to all its lines, the vertex of the pencil; in the two 
other types (the improper pencils ], this point is lacking. We 
shall now introduce, by convention, a fictitious entity, called an 
improper point, and regard it as pertaining to all the lines of 
the pencil. With this convention, every pencil has a vertex, 

The Complete Line and Plane. 22Q 

which will be a proper point, or an improper point, accord 
ing to the different cases. The hyperbolic plane, regarded 
as the aggregate of all its points, proper and improper, will 
be called the complete projective plane. 

3. The Complete Projective Line. The improper 
points are of two kinds. They may be the vertices of pen 
cils of parallels, or the vertices of pencils of non-intersecting 
lines. The points of the first species are obtained in the 
same way, and have the same use, as the points at infinity 
common to two Euclidean parallels. For this reason we shall 
call them points at infinity on the hyperbolic plane, when it 
is necessary to distinguish them from the others. The points 
of the second species will be called ideal points. 

It will be noticed that, while every line has only one 
point at infinity on the Euclidean plane, it has two points at 
infinity on the hyperbolic plane, there being two distinct 
directions of parallelism for each line. Also that, while the 
line on the Euclidean plane, with its point at infinity, is 
closed, the hyperbolic line, regarded as the aggregate of 
its proper points, and of its two points at infinity, is open. 
The hyperbolic line is closed by associating with it all the 
ideal points, which are common to it and to all the lines on 
the plane which do not intersect it. 

From this point of view we regard the line as made 
up of two segments, whose common extremities are the two 
points at infinity of the line. Of these segments, one contains, 
in addition to its ends, all the proper points of the line; the 
other all its improper points. The line, regarded as the 
aggregate of its points, proper and improper, will be called 
the complete projective line. 

4. Combination of Elements. We assume for the 
concrete representation of a point of the complete projective 

22Q App. IV- The Independ. of Proj. Geo. from Euclid s. Post. 

(i) its physical image, if it is a proper point; 

(ii) a line which passes through it, and the relative 
direction of the line, if it is a point at infinity; 

(iii) the common perpendicular to all the lines passing 
through it, if it is an ideal point. 

We shall denote a proper point by an ordinary capital 
letter; an improper point by a Greek capital; and to this 
we shall add, for an ideal point, the letter which will 
stand for the representative line of that point. Thus a point 
at infinity will be denoted, e. g., by Q, while the ideal point, 
through which all lines perpendicular to the line o pass, will 
be denoted by Q - 

On this understanding, if we make no distinction be 
tween proper points and improper points, not only can we 
affirm the unconditional validity of the projective postulate: 
two arbitrary lines have a common point: but we can also 
construct this point, understanding by this construction the 
process of obtaining its concrete representation. In fact, if the 
lines meet, in the ordinary sense of the term, or are parallel, 
the point can be at once obtained. If they are non-inter 
secting, it is sufficient to draw their common perpendicular, 
according to the rule obtained in Appendix III 4. 

On the other hand, we are not able to say that the 
second postulate of projective geometry two points determine 
a line and the corresponding constructions, are valid un 
conditionally. In fact no line passes through the ideal point 
Q and through the point at infinity Q on the line o, since 
there is no line which is at the same time parallel and per 
pendicular to a line o. 

Before indicating how we can remove this and other 
exceptions to the principle that a line can be determined by 
a pair of points, we shall enumerate all the cases in which 
two points fix a line, and the corresponding constructions: 

a) Two proper points. The line is constructed as usual. 

Combination of Elements. 23 I 

b) A proper point [0] and a point at infinity [Q]. The 
line OQ is constructed by drawing the parallel through to 
the line which contains Q, in the direction corresponding 
to Q. (Appendix III). 

(c) A proper point [0] and an ideal point [FJ]. The line 
Of f is constructed by dropping the perpendicular from to 
the line c. ,_j 

(d) Two points at infinity [Q, Q ]. The line QQ is the 
common parallel to the two lines bounding an angle, the 
construction for which is given in Appendix III 5. 

(e) An ideal point [FJ and a point at infinity [Q], not 
lying on the representative line c of the ideal point. The line 
Qf;. is the line which is parallel to the direction given by Q 
and perpendicular to c. The construction is given in Append 
ix El 6. 

(f) Two ideal points [l~ [~V], whose represmtative lines 
c, c do not intersect. The line I~ c rV, is constructed by drawing 
the common perpendicular to c and c (Appendix III 4). 

The pairs of points which do not determine a line are 
as follows: 

(i) an ideal point and a point at infinity, lying on the 
representative line of the ideal point; 

(ii) two ideal points, whose representative lines are 
parallel, or meet in a proper point. 

5. Improper Lines. To remove the exceptions men 
tioned above in (i) and (ii), new entities must be introduced. 
These we shall call improper lines, to distinguish them from 
the ordinary or proper lines. 

These improper lines are of two types: 

(i) If Q is a point at infinity, every line of the -pencil Q 
is the representative entity of an ideal point. The locus of 
these ideal points, together with the point Q, is an im 
proper line of the first type, or -line at infinity. It will be 
denoted by tu. 

232 ApP- Iv". The Independ. of Proj. Geo. from Euclid s Post. 

(ii) If A is a proper point, every line passing through A 
is the representative entity of an ideal point. The locus of 
these ideal points is an improper line of the second type, or 
ideal line. It will be denoted by a^. The proper point A 
can be taken as representative of the ideal line Q.A. 

These definitions of the terms line at infinity and ideal 
line allow us to state that two points, which do not belong 
to a proper line, determine either a line at infinity, or an 
ideal line. Hence, dropping the distinction between proper 
and improper elements, the projective postulate two points 
determine a line is universally true. 

We must now show that, with the addition of the im 
proper lines, any two lines have a common point. The 
various cases in which the two lines are proper have been 
already discussed ( 4). There remain to be examined the 
cases in which at least one of the lines is improper. 

(i) Let r be a proper line and uu an improper line, 
passing through the point Q at infinity. The point uur is the 
ideal point, which has the line passing through Q and per 
pendicular to r for representative line. 

(ii) Let r be a proper line and oc^ an ideal line. The 
point ra.A is the ideal point, which has the line passing 
through A and perpendicular to r for its representative line. 

(iii) Let o and u/ be two lines at infinity, to which 
belong the points Q and Q respectively. The point uuu/ is 
the ideal point, whose representative line is the line joining 
the points Q and Q . 

(iv) Let O.A, $B be two ideal lines. The point OLA$ is 
the ideal point, whose representative line is the line joining 
A and B. 

(v) Let uu and U.A be a line at infinity and an ideal line. 
The point wa^ is the ideal point, whose representative line 
is the line joining A to Q. 

Thus we have demonstrated that the two fundamental 

Use of Improper Elements. 333 

postulates of projective plane geometry hold on the hyper 
bolic plane. 

6. Complete Projective Space and the Validity of Pro- 
jtctive Geometry in the Hyperbolic Space. We can introduce 
improper points, lines and planes, into the Hyperbolic Space 
by the same method which has been followed in the preced 
ing paragraphs. We can then extend the fundamental pro 
positions of projective geometry to the complete projective 
space. Thereafter, following the lines laid down by STAUDT, 
all the important projective properties of figures can be de 
monstrated. Thus the validity of projective geometry in the 
LoBATSCHEWSKY-BoLYAi Space is established. 

7. Independence of Projective Geometry from the Fifth 
Postulate. Let us suppose that in a connected argument, 

founded on the group of postulates A, B , H, the only 

hypotheses which can be used are / M / 2 , /. Also that 

from the fundamental postulates and any one whatever of 
the I s t a certain proposition M can be derived. Then we 
may say that M is independent of the / s. 

It is precisely in this way that the independence of pro 
jective geometry from the Fifth Postulate is proved, since 
we have shown that it can be built up, starting from the 
group of postulates common to the three systems of geo 
metry, and then adding to them any one of the hypotheses 
on parallels. 

The demonstration of the independence of M from any 
one of the I s t founded on the deduction (cf. S 59) 

{A,,.. ff,I r } D ^ (r==If2f ...) 

may be called indirect, reserving the term direct demonstration 
for that which shows that it is possible to obtain M without 
introducing any of the J s at all. Such a possibility, from the 
theoretical point of view, is to be expected, since the 

234 ^PP ^ ^ ie Independ. of Proj. Geo. from Euclid s Post. 

preceding relations show that neither any single /, nor any 
group of them, is necessary to obtain M. If we wish to give 
a demonstration of the type 

(A, B, . . . ff] D M, 

in which the / s do not appear at all, we may meet difficult 
ies not always easily overcome, difficulties depending on the 
nature of the question, and on the methods we may adopt 
to solve it. So far as regards the independence of projective 
geometry from the Fifth Postulate, we possess two interesting 
types of direct proofs, founded on two different orders of 
ideas. One employs the method of analysis: the other that 
of synthesis. We shall now briefly describe the views on 
which they are founded. 

8. Bdtr ami s Direct Demonstration of the Independ 
ence of Projective Geometry from the Fifth Postulate. The 
demonstration implicitly contained in BELTRAMI S ^Saggio" of 
1868 must be placed first in chronological order. Referring 
to the Saggid , let us suppose that between the points of a 
surface F, (or of a suitably limited region of the surface), and 
the points "of an ordinary plane area, there can be established 
a one-one correspondence, such that the geodesies of the former 
are represented by the straight lines of the latter. Then, to the 
projective properties of plane figures, which express the 
collinearity of certain points, the concurrence of certain 
lines, etc., correspond similar properties of the correspond 
ing figures on the surface, which are deduced from the first, 
by simply changing the words plane and line into surface 
and geodesic. If all this is possible, we should naturally say 
that the projective properties of the -corresponding plane 
area are valid on the surface F\ or, more simply, that the 
ordinary projectivity of the plane holds upon the surface. 
We shall now put this result in an analytical form. 

Let u and v be the (curvilinear) coordinates of a point 

Beltrami s Direct Demonstration. 


on F, and x and y those of the representative point on the 
plane. The correspondence between the points (u, v) and 
(x, y) will be expressed analytically by putting 

where /and <p are suitable functions. 

To the equation 

\\) (u, v) = o 
of a geodesic on 7^, let us now apply the transformation (i). 

We must obtain a linear equation in x, y, since, by our 
hypothesis, the geodesies of F are represented by straight 
lines on the plane. 

But the equations (i) can also be interpreted as formulae 
giving a transformation of coordinates on F. We can there 
fore conclude that: ff, by a suitable choice of a system of 
curvilinear coordinates on the surface F, the geodesies of that 
surface can be represented by linear equations, the ordinary 
projective geometry is valid on the surface. 

Now BELTRAMI has shown in his Saggio* that on surfaces 
of constant curvature it is always possible to choose a system 
of coordinates (z/, z/), for which the general integral of the 
differential equation of the geodesies takes the form 
ax 4- by + c = o. 

Hence, from what has been said above, it follows that: 

Plane projective geometry is valid on the surfaces of con 
stant curvature with respect to their geodesies. 

But, according to the value of the curvature, the geo 
metry of these surfaces coincides with that of the Euclidean 
plane, or of the Non-Euclidean planes. 

It follows that: 

The method of Beltrami, applied to a plane on which are 
valid the metrical concepts common to the three geometries, leads 

236 App. IV. The Independ. of Proj. Geo. from Euclid s Post. 

to the foundation of plane projective geometry without the 
assumption of any hypothesis on parallels. 

This result and the argument we have employed in ob 
taining it are easily extended to space. BELTRAMI S memoir 
referring to this is the Teoria fondamentale degli spazii di 
curvatura costante, quoted in the note to 75. 

9. Klein s Direct Demonstration of the Independence 
of Protective Geometry from the Fifth Postulate. The method 
indicated above is not the only one which will serve our 
purpose. In fact, we might be asked if we could not construct 
projective geometry independently of any metrical consider 
ation; that is, starting from the notions of point, line, plane, 
and from the axioms of connection and order, and the prin 
ciple of continuity. 1 In 1871 KLEIN was convinced of the 
possibility of such a foundation, from the consideration of 
the method followed by STAUDT in the construction of his 
geometrical system. There remained one difficulty, relative 
to the improper points. STAUDT, following PONCELET, makes 
them to depend on the ordinary theory of parallels. To 
escape the various exceptions to the statement that two 
coplanar lines have a common point, due to the omission of 
the Euclidean hypothesis, KLEIN proposed to construct projective 
geometry in a limited (and convex) region of space, such, e. g., 
as that of the points inside a tetrahedron. With reference to 
such a region, for the end he has in view, every point on 
the faces of, or external to, the tetrahedron must be con 
sidered as non-existent. Also we must give the name of line 
and plane only to the portions of the line and plane belonging 
to the region considered. Then the graphical postulates of 
connection, order, etc., which are supposed true in the whole 

* For this nomenclature for the Axioms, cf. TOWNSEND S 
translation of HUMBERT S Foundations of Geometry, p. I (Open Court 
Publishing Co. 1902). 

Klein s Direct Demonstration. 


of space, are verified in the interior of the tetrahedron. Thus 
to construct projective geometry in this region, it is neces 
sary, with suitable conventions, that the propositions on the 
concurrence of lines, etc. should hold without exception. 
These are not always true, when the word point means 
simply point inside the tetrahedron. 

KLEIN showed briefly, while various later writers dis 
cussed the question more fully, how the space inside the 
tetrahedron can be completed by fictitious entities, called 
ideal points, lines and planes, so that when no distinction 
is made between the proper entities (inside the tetrahedron) 
and the ideal entities, the graphical properties of space, on 
which all projective geometry is constructed, are completely 

From this there readily follows the independence of 
projective geometry from EUCLID S Fifth Postulate. 

Appendix V. 

The Impossibility of Proving Euclid s 
Parallel Postulate. 1 

An Elementary Demonstration of this Impossibility founded 
upon the Properties of the System of Circles orthogonal to a 
Fixed Circle. 

I. In the concluding article ( 94) various arguments 
are mentioned, any one of which establishes the independence 
of EUCLID S Parallel Postulate from the other assumptions on 
which Euclidean Geometry is based. One of these has been 
discussed in greater detail in Appendix IV. In the articles 
which follow there will be found another and a more ele 
mentary proof that the BOLYAI-LOBATSCHEWSKY system of 
Non-Euclidean Geometry cannot lead to any contradictory 
results, and that it is therefore impossible to prove EUCLID S 
Postulate or any of its equivalents. This proof depends, for 
solid geometry, upon the properties of the system of spheres all 
orthogonal to a fixed sphere, while for plane geometry the 
system of circles all orthogonal to a fixed circle is taken. 
In the course of the discussion many of the results of Hyper 
bolic Geometry are deduced from the properties of this 
system of circles. 

1 This Appendix, added to the English translation, is based 
upon WELLSTEIN S work, referred to on p. 180, and the following 
paper by CARSLAW; The Bolyai-Lobatschewsky Non-Euclidean Geo 
metry : an Elementary Interpretation of this Geometry and some Results 
which follow from this Interpretation, Proc. Edin. Math. Soc. Vol. 
XXVIII, p. 95 (1910). 

Cf. also : J. WELLSTEIN, Zusammenhang zwischen zwei cuklid- 
ischen Bildern der nichteuklidischen Geometrie. Archiv der Math. u. 
Physik (3). XVII, p. 195 (1910). 

Ideal Lines. 


The System of Circles passing through a fixed Point. 

2. We shall examine first of all the representation of 
ordinary Euclidean Geometry by the geometry of the system 
of spheres all passing through a fixed point. In plane geo 
metry this reduces to the system of circles through a fixed 
point, and we shall begin with that case. 

Since the system of circles through a point O is the 
inverse of the system of straight lines lying in the plane, to 
every circle there corresponds a straight line, and the circles 
intersect at the same angle as the corresponding lines. The 
properties of the set of circles could be established from the 
knowledge of the geometry of the straight lines, and every 
proposition concerning points and straight lines in the one 
geometry could at once be interpreted as a proposition con 
cerning points and circles in the other. 

There is another way in which the geometry of these 
circles can be established independently. We shall first de 
scribe this method, and weshall then see that from this inter 
pretation of the Euclidean Geometry we can easily pass to a 
corresponding representation of the Non-Euclidean Geometry. 

3. Ideal Lines. 

It will be convenient to speak ot the plane, of the 
straight lines and the plane of the circles, as two separate 
planes. We have seen that to every straight line in the plane 
of the straight lines, there corresponds a circle in the plane 
of the circles. We shall call these circles Ideal Lines. The 
Ideal Points will be the same as ordinary points, except that 
the point O will be excluded from the domain of the Ideal 

On this understanding we can say that Any two different 
Ideal Points, A, B, determine the Ideal Line AB; just as, in 
Euclidean Geometry, any two different points A, B deter 
mine the straight line AB. 

240 Appendix V. Impossibility of proving Euclid s Postulate. 

As the angle between the circles in the one plane is 
equal to the angle between the corresponding straight lines 
in the other, we define the angle between two Ideal Lines as 
the angle between the corresponding straight /mes. Thus we 
can speak of Ideal Lines being perpendicular to each other, 
or cutting at any angle. 

4. Ideal Parallel Lines. 

Let BC (cf. Fig. 83) be any straight line and A a point 
not lying upon it. 

Let AM be the perpendicular to BC, and AM lt AM 2 , 
3 , . . . different positions of the line AM, as it revolves 
from the perpendicular position through two right angles. 

The lines begin by cutting BC on the one side of M, 
and there is one line separating the lines which intersect 
BC on the one side, from those which intersect it on the 
other. This line is the parallel through A to BC. 

In the corresponding figure for the Ideal Lines (cf. 
Fig. 84), we have the Ideal Line through A perpendicular to 
the Ideal Line BC; and the circle which passes through A, 
and touches the circle OBC at O, separates the circles 
through A, which cut BC on the one side of M, from those 
which cut it on the other. 

Ideal Parallels. 

2 4 I 

We are thus led to define Parallel Ideal Lines as follows: 

The Ideal Line through any point parallel to a given 

Ideal Line is the circle of the system which touches at O\the 

circle coinciding with the given line and also passes through the 

given point. 

Fig. 84. 

Thus any two circles of the system which touch each 
other at O will be Ideal Parallel Lines. Two Ideal Lines, 
which are each parallel to a third Ideal Line, are parallel to 
each other, etc. 

5. Ideal Lengths. 

Since EUCLID S Parallel Postulate is equivalent to the 
assumption that one, and only one, straight line can be 
drawn through a point parallel to another straight line, and 
since this postulate is obviously satisfied by the Ideal Line, 


242 Appendix V. Impossibility of proving Euclid s Postulate. 

in the geometry of these lines, EUCLID S Theory of Parallels 
will be true. 

But such a geometry will require a measurement of 
length. We must now define what is meant by the Ideal 
Length of an Ideal Segment. In other words we must define 
the Ideal Distance between two points. It is clear that if the 
two geometries are to be identical two Ideal Segments must 
be regarded as of equal length, when the corresponding 
rectilinear segments are equal. We thus define the Ideal 
Length of an Ideal Segment as the length of the rectilinear 
segment to which it corresponds. 

It will be seen that the Ideal Distance between two 
points A, B is such that, if C is any other point on the 

distance AB = distance AC -\- distance CB. 

The other requisite for distance is that it is unaltered 
by displacement, and when we come to define Ideal Dis 
placement we shall have to make sure that this condition is 
also satisfied. 

It is clear that on this understanding the Ideal Length 
of an Ideal Line is infinite. If we take equal steps along 
the Ideal Line BC from the foot of the perpendicular (cf. 

Fig. 84) the actual lengths of the arcs MM* , M^M 2 , etc , 

the Ideal Lengths of which are equal, become gradually 
smaller and smaller, as we proceed along the line towards O. 
It will take an infinite number of such steps to reach O, just 
as it will take an infinite number of steps along BC from M 
(cf. Fig. 83) to reach the point at which BC\s> met by the 
parallel through A. We have already seen that the domain 
of Ideal Points contains all the points of the plane except 
O. This was required so that the Ideal Line might always 
be determined by two different points. It is also needed for 
the idea of bet\veen-ness . On the straight line AB we can 
say that C lies between line A and B if, as we proceed along 

Ideal Lengths. 


AB from A to B, we pass through C. On the Ideal Line AB 
(cf. Fig. 85) the points C t and C 2 would both lie between 
A and B, unless the point O were excluded. In other words 
this convention must be made so that the Axioms of Order x 
may appear in the geometry of the Ideal Points and Lines. 

Fig 85. 

On this understanding, and still speaking of plane geo 
metry, we can say that two Ideal Lines are parallel when they 
do not meet, however far they are produced. 

To obtain an expression for the Ideal Length of an 
Ideal Segment we may take the radius of inversion k to 
be unity. 

Consider the segment AB and the rectilinear segment 
dp to which it corresponds. Then we have (Fig. 86) 
ap Op Op . OP, J& 
AB ~~ OA "* OA. OB ~ OA. OB 

See Note on p. 236. 

1 6* 

244 ApP en dix V. Impossibility of proving Euclid s Postulate. 

Hence we define the Ideal Length of the segment AB as 

OA . OB 

We shall now show that the Ideal Length of an Ideal 
Segment is unaltered by inversion with regard to any circle of 
the svstem. 

Fig. 86. 

Let OD be any circle of the system and let C be its 
centre (Fig. 87). 

Then inversion changes an Ideal Line into an Ideal 

Let the Ideal Segment AB invert into the Ideal Segment 
A B . These two IdeaPLines intersect at the point >, where 
the circle of inversion]meets AB. 


the Ideal Length of AD AD i A D 

the Ideal 

j^engin 01 /^t/ sju i /* j.j 

Length of A D = ~ OA . OD/~OA . OD 

~A r b OA 

But from the triangles CAD, CAD and OAC, OA C, 

we find 

Ideal Displacements. 


A ~D 



A O 

Thus the Ideal Length of AD = the Ideal Length olA D. 
Similarly we find BD and B D have the same Ideal Length, 
and therefore AB and A B have the same Ideal Length. 

6. Ideal Displacements. 

The length of a segment must be unaltered by dis 
placement. This leads us to consider the definition of Ideal 
Displacement. Any displacement may be produced by re 
peated applications of reflection; that is, by taking the image 
of the figure in a line (or in a plane, in the case of solid 
geometry). For example, to translate the segment AB (cf. 
Fig. 88) into another position on the same straight line, we 

246 Appendix V. Impossibility of proving Euclid s Postulate. 

may reflect the figure, first about a line perpendicular to and 
bisecting BB , and then another reflection about the middle 
point of AB would bring the ends into their former positions 
relative to each other. Also to move the segment AB into 



Fig. 88. 

the position AB (cf. Fig. 89) we can first take the image ot 
AB in the line bisecting the angle between AB and AB , 
and then translate the segment along AB to its final 

We proceed to show 
that inversion about any 
circle of the system is 
equivalent to reflection of 
the Ideal Points and Lines 
in the Ideal Line which 
coincides with the circle 
of inversion. 

B f Let C (Fig. 90) be 

the centre of any circle 
of the system, and let A 
be the inverse of any 
point A with regard to 
this circle. Then the 
circle OAA is orthogonal 
to the circle of inversion. 
In other words, such inversion changes any point A into a 
point A on the Ideal Line perpendicular to the circle of in 
version. Also the Ideal Line AA is bisected by that circle 
at J/, since the Ideal Segment AM inverts into the segment 
AM, and Ideal Lengths are unaltered by such inversion. 

Again let AB be any Ideal Segment, and by inversion 

Fig. 89. 

Ideal Reflection. 

2 47 

with regard to any circle of the system let it take up the 
position A # (Fig. 87). We have seen that the Ideal Length 
of the segment is unaltered: and it is clear that the two 
segments, when produced, meet on the circle of inversion, 
and make equal angles with it. Also the Ideal Lines A A 

Fig. 90. 

and BI? are perpendicular to, and bisected by, the Ideal 
Line with which the circle of inversion coincides. 

Such an inversion is, therefore, the same as reflection, 
and translation will occur as a special case of the above, 
when the circle of inversion is orthogonal to the given 
Ideal Line. 

We thus define Ideal Reflection in an Ideal Line as in 
version with this line as the circle of inversion. 

It is unnecessary to say more about Ideal Displace 
ments than that they will be the result of Ideal Reflection. 

With these definitions it is now possible to translate 
every proposition in the ordinary plane geometry into a 

248 Appendix V. Impossibility of proving Euclid s Postulate. 

corresponding proposition in this Ideal Geometry. We have 
only to use the words Ideal Points, Lines, Parallels, etc., 
instead of the ordinary points, lines, parallels, etc. The 
argument employed in proving a theorem, or the con 
struction used in solving a problem, will be applicable, 
word for word, in the one geometry as well as in the other, 
for the elements involved satisfy the same laws. This is the 
dictionary method so frequently adopted in the previous 
pages of this book. 

7. Extension to Solid Geometry. The System of 
Spheres passing through a fixed point. 

These methods may be extended to solid geometry. In 
this case the inversion of the system of points, lines, and 
planes gives rise to the system of points, circles intersecting 
in the centre of inversion, and spheres also intersecting in 
that point. The geometry of this system of spheres could be 
derived from that of the system of points, lines and planes, 
by interpreting each proposition in terms of the inverse 
figures. For our purpose it is better to regard it as derived 
from the former by the invention of the terms: Ideal 
Point, Ideal Line, Ideal Plane, Ideal Length and Ideal Dis 

The Ideal Point is the same as the ordinary point, but 
the point O is excluded from the domain of Ideal Points. 

The Ideal Line through two Ideal Points is the circle of 
the system which passes through these two points. 

The Ideal Platie through three Ideal Points, not on an 
Ideal Line, is the sphere of the system which passes through 
these three points. 

Thus the plane geometry, discussed in the preceding 
articles, is a special case of this plane geometry. 

Ideal Parallel Lines are denned as before. The line 
through A parallel to BC is the circle of the system, lying 

Extension to Solid Geometry. 249 

on the sphere through O, A, JB y and C, which touches the 
circle given by the Ideal Line BC at O and passes through A. 

It is clear that an Ideal Line is determined by two 
points, as a straight line is determined by two points. An 
Ideal Plane is determined by three points, not on an Ideal 
Line, as an ordinary plane is determined by three points, 
not on a straight line. If two points of an Ideal Line lie on 
an Ideal Plane, all the points of the line do so: just as if two 
points of a straight line lie on a plane, all its points do so. 
The intersection of two Ideal Planes is an Ideal Line; just as 
the intersection of two ordinary planes is a straight line. 

The measurement of angles in the two spaces is the same. 

For the measurement of length we adopt the same de 
finition of Ideal Length as in the case of two dimensions. 
The Ideal Length of an Ideal Segment is the length of the 
rectilinear segment to which it corresponds. To these defi 
nitions it only remains to add that of Ideal Displacement. 
As in the two dimensional case, this is reached by means of 
Ideal Reflection : and it can easily be shown that if the system 
of Ideal Points, Lines and Planes is inverted with regard to 
one of its spheres, the result is equivalent to a reflection of the 
system in this Ideal Plane. 

This Ideal Geometry is identical with the ordinary 
Euclidean Geometry. Its elements satisfy the same laws: 
every proposition valid in the one is also valid in the other: 
and from the results of Euclidean Geometry those of the 
Ideal Geometry can be inferred. 

In the articles that follow we shall establish an Ideal 
Geometry whose elements satisfy the axioms upon which the 
Non-Euclidean Geometry of BOLYAI-LOBATSCHEWSKY is based. 
The points, lines and planes of this geometry will be figures 
of the Euclidean Geometry, and from the known properties 
of these figures, we could state what the corresponding the 
orems of this Non-Euclidean Geometry would be. Also from 

250 Appendix V. Impossibility of proving Euclid s Postulate. 

some of its constructions, the Non-Euclidean constructions 
could be obtained. This process would be the converse of 
that referred to in dealing with the Ideal Geometry of the 
preceding articles: since, in that case, we obtained the the 
orems of the Ideal Geometry from the corresponding Eu 
clidean theorems. 

The Geometry of the System of Circles Orthogonal 
to a Fixed Circle. 

8. Ideal Points, Ideal Lines and Ideal Parallels. 

In the Ideal Geometry discussed in the previous articles, 
the Ideal Point was the same as the ordinary point, and the 
Ideal Lines and Planes had so far the characteristics of 
straight lines and planes that they were lines and surfaces 
respectively. Geometries can be constructed in which the 
Ideal Points, Lines and Planes are quite removed from 
ordinary points, lines, and planes: so that the Ideal Points 
no longer have the characteristic of having no parts: and 
the Ideal Lines no longer boast only length, etc. What is 
required in each geometry is that the entities concerned 
satisfy the axioms which form the foundations of geometry. 
If they satisfy the axioms of Euclidean Geometry, the argu 
ments, which lead to the theorems of that geometry, will 
give corresponding theorems in the Ideal Geometry: and if 
they satisfy the axioms of any of the Non-Euclidean Geom 
etries, the arguments, which lead to theorems in that Non- 
Euclidean Geometry, will lead equally to theorems in the 
corresponding Ideal Geometry. 

We proceed to discuss the geometry of the system of 
circles orthogonal to a fixed circle. 

Let the fundamental circle be of radius k and centre O. 

Let A , A" be any two inverse points, A being inside 
the circle. Every such pair of points (A , A"), is an Ideal 
Point (A} of the Ideal Geometry with which we shall now deal. 

Circles orthogonal to a fixed Circle. 


If two such pairs of points are given that is, two Ideal 
Points (A, ), (Fig. 92) these determine a circle which is 
orthogonal to the fundamental circle. Every such circle is 
an Ideal Line of this Ideal Geometry. 


Fig. 91. 

Hence any two different Ideal Points determine an Ideal 
Line. In the case of the system of circles passing through a 
fixed point O, this point O was excluded from the domain 
of the Ideal Points. In this system of circles all orthogonal 
to the fundamental circle, the coincident pairs of points lying 
on the circumference of that circle are excluded from the 
domain of the Ideal Points. 

We define the angle between two Ideal Lines as the angle 
between the circles which coincide with these lines. 

We have now to consider in what way it will be proper 
to define Parallel Ideal Lines. 

Let AM be the Ideal Line through A, perpendicular to 
the Ideal Line BC\ in other words, the circle of the system 
passing through A\ A , and orthogonal to the circle through 
#, ", C and C" (cf. Fig. 92). 

2C2 Appendix V. Impossibilty of proving Euclid s Postulate. 

Imagine AM to rotate about A so that those Ideal 
Lines through A cut the Ideal Line BC at a gradually 
decreasing angle. The circles through A which touch the given 


Fig. 92. 

circle BC at the points U, V> where it meets the fundamental 
circle, are Ideal Lines of the system. They separate the 
lines of the pencil of Ideal Lines through A : which cut the 
Ideal Line BC, from those which do not cut that line. All 
the lines in the angle (p, shaded in the figure, do not cut 
the line BC\ all those in the angle ip, not shaded, do cut 
this line. This property is exactly what is assumed in the 
Parallel Postulate upon which the Non-Euclidean Geometry 
of BoLYAi-LoBATSCHEWSKY is based. We therefore are led to 
define Parallel Ideal Lines in this Plane Ideal Geometry as 
follows : 

The Ideal Lines through an Ideal Point parallel to a 
given Ideal Line are the two circles of the system passing 

Some Theorems in this Geometry. 253 

through the given point, which touch the circle with which the 
given line coincides at the points where it meets the fundam 
ental circle. 

Thus we have in this Ideal Geometry two parallels 
through a point to a given line: a right-handed parallel, and 
a left-handed parallel: and these separate the lines of the 
pencil which intersect the given line from those which do 
not intersect it. 

Some Theorems of this Non-Euclidean Geometry. 

9. At this stage we can say that any of the theorems 
of the BOLYAI-LOBATSCHEWSKY Non-Euclidean Geometry, in 
volving angle properties only, will hold in this Ideal Geo 
metry and vice versa. Those involving lengths we cannot yet 
discuss, as we have not yet denned Ideal Lengths. For 
example, it is obvious that there are triangles in which all 
the angles are zero (cf. Fig. 93). The sides of such triangles 
are parallel in pairs. Thus the sum of the angles of an Ideal 
Triangle is certainly not always equal to two right angles. 
We can prove that this sum is always less than two right 
angles by a simple application of inversion, as follows: 

Let C,, C 2 , C 3 be three circles of the system, forming 
an Ideal Triangle. Invert these circles from the point of 
intersection / of Cj. and C 2 , which lies inside the fundament 
al circle. Then C x and C 2 become two straight lines d 
and C 2 through /. Also the fundamental circle C inverts 
into a circle C cutting */ and C 2 at right-angles, so that 
its centre is /. Again, the circle C 3 inverts into a circle C/, 
cutting C at right-angles. Hence its centre lies outside C . 
We thus obtain a triangle , in which the sum of the angles 
is less than two right-angles, and since these angles are equal 
to the angles of the Ideal Triangle, this result holds also for 
the Ideal Triangle. 

2 EJ4 Appendix V. Impossibility of proving Euclid s Postulate. 

Finally, it can be shown that there is always one, and 
only one, circle of the system cutting two non-intersecting 
circles of the system at right-angles. In other words, two 

Fig. 93- 

non-intersecting Ideal Lines have a common perpendicular. 
All these results must be true in the Hyperbolic Geometry. 

10. Ideal Lengths and Ideal Displacements. 

Before we can proceed to the discussion of the metrical 
properties of this geometry, we must define the Ideal Length 
of an Ideal Segment. It is clear that this must be such that 
it will be unaltered, if we take the points A \ B", as defining 
the segment AB, instead of the points A , B . It must make 
the complete line infinite in length. It must satisfy the distri 
butive law distance AB = distance AC + distance CB, 

Ideal Lengths. 


if C is any other point on the segment AB, and it must 
also remain unaltered by Ideal Displacement. 

We define the Ideal Length of any segment AB as 

(A V I BT\ 
10 \A U\ WU) 

where U y V are the points where the Ideal Line AB meets the 
fundamental circle (cf. Fig. 91). 


Fig. 94. 

This expression obviously involves the Anharmonic 
Ratio of the points UAB V. It will be seen that this de 
finition satisfies the first three of the conditions named above. 
It remains for us to examine what must represent dis 
placement in this Ideal Geometry. 

Let us consider what is the effect of inversion with 
regard to a circle of the system upon the system of Ideal 
Points and Lines. 

Let A A" be any Ideal Point A (cf. Fig. 94). Let the 

2 $ 6 Appendix V. Impossibility of proving Euclid s Postulate. 

circle of inversion meet the fundamental circle in C, and let 
D be its centre. Let A , A" invert into B , B" . Since the 
circle A A C touches the circle of inversion at C, its inverse 
also touches that circle at C. But a circle passes through 
A , A", B and ", and the radical axes of the three circles 

A A"C, B "C, AA B B" 
are concurrent. 

Hence B" passes through O, and OB . OB" = OC 2 . 

Therefore inversion with regard to any circle of the 
system changes an Ideal Point into an Ideal Point. 

But it is clear that the circle A A B B" is orthogonal to 
the fundamental circle, and also to the circle of inversion. 

Thus the Ideal Line joining the Ideal Point A and the 
Ideal Point B, into which it is changed by this inversion, is 
perpendicular to the Ideal Line coinciding with the circle of 

We shall now prove that it is bisected by that Ideal 

Let the circle through AB meet the circle of inversion 
at M, and the fundamental circle in U and K It is clear 
that U and V are inverse points with regard to the circle of 
inversion [cf. Fig. 95]. 

Then we have: 

&Y _ CV 
CA 1 



A V B V CV^ CV2 



A V I M V M V I B V 

A U M U 

M V I B V 
M U\ B U 

Ideal Reflection. 


Hence the Ideal Length of AM is equal to the Ideal 
Length of MB. 

Thus we have the following result: 

Inversion with regard to a circle of the system changes 
any Ideal Point A into an Ideal Point B, such that the Ideal 
Line AB is perpendicular to, and i bisected ) by, the Ideal Line 
coinciding with the circle of inversion. 

Fi*. 95 

In other words, inversion with regard to such a circle 
causes any Ideal Point A to take the position of its image in 
the corresponding Ideal Line. 

We proceed to examine what effect such inversion has 
upon an Ideal Line. 

Since a circle, orthogonal to the fundamental circle, 


2C8 Appendix V. Impossibility of proving Euclid s Postulate. 

inverts into a circle also orthogonal to the fundamental circle, 
any Ideal Line AB inverts into another Ideal Line ab, pass 
ing through the point M, where AB meets the circle of in 
version (cf. Fig. 96). Also the points 7, V invert into the 

points u and v on the fundamental circle; and the lines AB 
and ab are equally inclined to the circle of inversion. 

It is easy to show that the Ideal Lengths of AM and 
BM are equal to those of a M and bM respectively, and it 
follows that the Ideal Length of the segment AB is unaltered 
by this inversion. Also we have seen that Aa and Bb are 
perpendicular to, and bisected by, the Ideal Line coinciding 
with this circle. 

// follows from these results that inversion with regard 
to any circle of the system has the same effect upon an Ideal 
Segment as reflection in the corresponding Ideal Line. 

We are thus again able to define Ideal Reflection in any 
Ideal Line as the inversion of the system of Ideal Points and 

Ideal Displacement. 


Lines with regard to the circle which coincides with this 
Ideal Line. 

It is unnecessary to define Ideal Displacements, as any 
displacement can be obtained by a series of reflections and 
any Ideal Displacement by a series of Ideal Reflections. 

We notice that the definition of the Ideal Length of 
any Segment fixes the Ideal Unit of Length. We may take 
this on one of the diameters of the fundamental circle, since 
these lines are also Ideal Lines of the system. Let it be the 
segment OP (Fig. 97). 

Fig. 97- 

Then we must have 

/ov\ rv\ 


i. e. 





and the point P divides the diameter in the ratio e : i. 

The Unit Segment is thus fixed for any position in the 


260 Appendix V. Impossibility of proving Euclid s Postulate. 

domain of the Ideal Points, since the segment OP can be 
moved so that one of its ends coincides with any given 
Ideal Point. 

A different expression for the Ideal Length 

SAV I Br\ 


would simply mean an alteration in the unit, and taking 
logarithms to any other base than e would have the same 

ir. Some further Theorems in this Non- Euclidean 

We are now in a position to establish some further 
theorems of the Hyperbolic Geometry using the metrical 
properties of this Ideal Geometry. 

In the first place we can state that Similar Triangles 
are impossible in this geometry. 

We also see that Parallel Ideal Lines are asymptotic; 
that is, these lines continually approach each other and the 
distance between them tends to zero. 

Further, it is obvious that as the point A moves away 
along the perpendicular MA to the line BC (cf. Fig. 92), the 
angle of parallelism diminishes from to zero in the limit. 

Again, we can prove from the Ideal Geometry that the 
Angle of Parallelism TT (/), corresponding to a segment /, is 
given by 

tan H (/) _ -P 

Consider an Ideal Line and the Ideal Parallel to it 
through a point A. 

Let AM (Fig. 98) be the perpendicular to the given line 
MU, and A U the parallel. 

Let the figure be inverted from the point J/", the radius of 
inversion being the tangent from M" to the fundamental circle. 

Further Theorems in this Geometry. 


Then we obtain a new figure (cf. Fig. 99) in which the 
corresponding Ideal Lengths are the same, since the circle 
of inversion is a circle of the system. The lines A M and 
MU become straight lines through the centre of the fund- 


Fig. 98- 

amental circle, which is the inverse of the point M . 
Also the circle A U becomes the circle au, touching the 
radius mu at ?/, and cutting ma at an angle TT (/). These 
radii, m u, m b, are also Ideal Lines of the system. 

The Ideal Length of the Segment AM is taken as /. 
(A B \M D\ 

t- l< *(*c\Wc) 


But at == k - k tan - 
and ,* = * + ^ta 

262 Appendix V. Impossibility of proving Euclid s Postulate, 
where k is the radius of the fundamental circle. 

COt ; 

Thus p 

and/ -tan^ 

Fig. 99. 

Finally, in this geometry there will be three kinds of 
circles. There will be the circle, with its centre at a finite 
distance; the Limiting Curve or Horocycle, with its centre at 
infinity, (at a point where two parallels meet) ; and the Equi 
distant Curve, with its centre at the imaginary point of inter 
section of two lines with a common perpendicular. 

The first of these curves would be traced out in the 
Ideal Geometry by one end of an Ideal Segment, when it is 
reflected in the lines passing through the other end; that is, 
by the rotation of this Ideal Segment about that end. The 
second occurs when the Ideal Segment is reflected in the 
successive lines of the pencil of Ideal Lines all parallel to it 
in the same direction; and the third, when the reflection 

Application to Euclid s Parallel Postulate. 263 

takes place in the system of Ideal Lines which all have a 
perpendicular with this segment. That these correspond to 
the common Circle, the Horocycle and the Equidistant Curve 
of the Hyperbolic Geometry is easily proved. 

12. The Impossibility of Proving Euc lid s Parallel 

We could obtain other results of the Hyperbolic Geo 
metry, and find some of its constructions, by further examin 
ation of the properties of this set of circles; but this is not 
our object. Our argument was directed to proving, by reas 
oning involving only elementary geometry, that it is impossible 
for any inconsistency or contradiction to arise in this Non- 
Euclidean Geometry. If such contradiction entered into this 
Plane Geometry, it would also occur in the interpretation of 
the result in the Ideal Geometry. Thus the contradiction 
would also be found in the Euclidean Geometry. We can, 
therefore, state that it is impossible that any logical incon 
sistency could be traced in the Plane Hyperbolic Geometry. It 
could still be argued that such contradiction might be found 
in the Solid Hyperbolic Geometry. An answer to this ob 
jection is at once forthcoming. The geometry of the system 
of circles, all orthogonal to a fixed circle, can be at once 
extended into a three dimensional system. The Ideal Points 
are taken as the pairs of points inverse to a fixed sphere, 
excluding the points on the surface of the sphere from their 
domain. The Ideal Lines are the circles through two Ideal 
Points. The Ideal Planes are the spheres through three Ideal 
Points, not lying on an Ideal Line. The ordinary plane enters 
as a particular case of these Ideal Planes, and so the Plane 
Geometry just discussed is a special case of a plane geo 
metry on this system. With suitable definitions of Ideal 
Lengths, Ideal Parallels and Ideal Displacements, we have 
a Solid Geometry exactly analogous to the Hyperbolic Solid 

264 Appendix V. Impossibility of proving Euclid s Postulate. 

Geometry. It follows that no logical inconsistency can exist 
in the Hyperbolic Solid Geometry, since if there were such 
a contradiction, it would also be found in the interpretation 
of the result in this Ideal Geometry; and therefore it would 
enter into the Euclidean Geometry. 

By this result our argument is complete. However far 
the Hyperbolic Geometry were developed, no contradictory 
results could be obtained. This system is thus logically 
possible; and the axioms upon which it is founded are not 
contradictory. Hence it is impossible to prove Euclid s 
Parallel Postulate, since its proof would involve the denial 
of the Parallel Postulate of BOLYAI-LOBATSCHEWSKY. 

Index of Authors. 

[The numbers refer to pages.] 

Aganis, (6 h Century). 8 n. 
Al-Nirizi, (9* Century). 7, 9. 
Andrade, J. 181, 194. 
Archimedes, (287212). 9, n, 
23, 25, 30, 34, 37, 46, 56, 59, 
119121, 144, 181, 183. 
Aristotle, (384322). 4, 8, 18, 19. 
Arnauld, (16121694). 17. 
Baltzer,R. (1818 1887). 1213. 
Barozzi, F. (16^ Century). 12. 
Bartels, J. M. C. (17691836). 

84, 912. 
Battaglini, G. (18261894). 86, 

100, 122, 1267. 
Beltrami, E. (18351900). 44, 

122, 1267, ^33. 1389. 145> 

147, ^73 S 2 346. 
Bernoulli, D. (17001782). 192. 
Bernoulli, J. (17441807). 44. 
Bessel, F. W. (17841846). 65 

-6 7 . 

Besthorn, R. O. 7. 
Bianchi, L. 129, 135, 209. 
Biot, J. B. (17741862). 52. 
Boccardini, G. 44. 
Bolyai, J. (18021860). 51, 6 1, ; 

65, 74, 96107, 109-116; 

1216, 128, 137, 141, 145. 147, 

152, 154, 1578, 161, 164, 

170, 173 5, 1778, 1934, 

200, 222, 225, 233, 238, 249, 

2523, 264. 

Bolyai, W. (1775 1856). 55, 60 

i, 656, 74, 96. 98101, 

120, 125 6. 
Boncampagni, B. (18211894). 

Bonola, R. (18751611) 15, 26, 

30, 115, 1767, 220. 
Borelli, G. A. (16081679). n, 

13, 17- 

Boy, W. 149. 

Campanus, G. (l3 th Century). 17. 
Candalla, F. (15021594). 17. 
Carnot, L. N.M. (17531823). 53. 
Carslaw, H. S. 40, 238. 
Cassani, P. (1832 1905). 127. 
Castillon, G. (1708-1791). 12. 
Cataldi, P. A. (1548? 1626). 13. 
Cauchy, A.L. (17891857)- 1 99- 
Cayley, A. (1821-1895). 127, 

148, 156, 1634, 174, 179- 
Chasles, M. (17931880). 155. 
Clavio, C. (15371612). 13, 17. 
Clebsch, A. (18331872). 161. 
Clifford, \V.K.( 1845 1879). 139, 

142, 200215. 

Codazzi, D. (18241873). 137. 
Commandino, F. (15091575). 

12, 17. 

Coolidge, J. L. 129. 
Couturat, L. 54. 
Cremona, L. (18301903). 123, 



Index of Authors. 

Curtze, M. (1837 1903). 7. 
D Alembert, J. le R. (1717 

1783). 52, 54, 192, 197-8. 
Dedekind, J.W. R. (1831 1899). 


Dehn, M. 30, 120, 144. 
Delambre, J. B. J. (17491822). 


De Morgan, A. (18061870). 52. 
Dickstein, S. 139. 
Duhem, P. 182. 
Eckwehr, J. W. v. (1789-1857). 

Engel, F. 16, 445, 5, 6o 6 4> 

66, 836, 88, 923, 96, 101, 

216 7, 220. 

Enriques, F. 156, 166, 183, 225. 

Eotvos, 125. 

Euclid (circa 330 275). I 8, 10, 
1214, 1620, 22, 38, 512, 
54-5, 61 2, 68, 75, 82, 85, 
92, 95, loi 2, 104, no, 112, 

118120, 127, 139, 141, 147, 
152, 1545, 157, 164, 176 

181, 183, 191 5, 199201, 
227, 2379, 241, 267. 

Fano, G. 153. 

Flauti, V. (17821863) 12. 

Fleischer, H. 156. 

Flye St. Marie, 91. 

Foncenex, D. de, (1 743 1 799). 
53, 146, 1902, 1978. 

Forti, A. (1818 ). 122, 1245. 

Fourier, J.B. (1768 1830). 545. 

Frattini, G. 127. 

Frankland, W. B. 2, 63. 

Friedlein, G. 2. 

Frischauf, J. 100, 126. 

Gauss, C. F. (17771855). 16, 
6068, 7078, 834, 86, 88, 
902, 99101, no, 113, 122 

-3, 127, 131, 135, 152, 177, 


Geminus, (ist. Century, B. C.). 3, 

7, 20. 
Genocchi, A. (1817 1889). 145, 

191, 1989. 
Gerling, Ch. L. (17881864). 

65, 66, 76-7, I2i 2. 
Gherardo daCremona, (i2*h Cen 
tury). 7. 
Giordano Vitale, (1633 1711). 14 

-15, 17, 26. 
Goursat, E. 145. 
Gregory, D. (16611710) 17, 20. 
Grossman, M. 169, 225. 
Giinther, S. 127. 
Halsted, G. B. 44, 86, 93, 100, 139. 
Hauff, J. K. F. (17661846). 75. 
Heath, T. L. i, 2, 63. 
Heiberg, J. L. 1, 2, 7, 181. 
Heilbronner, J. C. (1706 1745). 

Helmholtz, H. v. (1821 1894). 

126, 145, 152-3, 176-7, 179. 
Hilbert, D. 23, 1456, 2224, 

Hindenburg, K. F. (1741 1808). 


Hoffman, J. (17771866). 12. 
Holmgren, E. A. 145 6. 
Hoiiel, J. (18231866). 52, 86, 

ico, 121, 1237, 139,147, 152. 
Kant, I. (1724 1804). 64, 92, 121. 
Kastner, A. G. (17191800). 50, 

60, 64, 66. 
Killing, W. 91, 215. 
Klein, K. F. 129, 138, 148, 153, 

158, 164, 176, 180, 200, 211, 
213-5. 236-7. 

Kltigel, G. S. (17391812). 12, 
44, 5i 64, 77, 92. 

Index of Authors. 

Kiirschak, J. 113. 

Lagrange, J. L. (17361813). 53 

4, 1823, 198- 
Laguerre, E. N. (18341866). 


Lambert, J. H. (17281777). 44 
5J, 58, 65-6, 74, 77-8, 
81 2, 92, 97, 107, 129, 139, 

Laplace, P. S. (1749-1827), 53 

-54, 198- 
Legendre, A. M. (17521833), 

29, 44, 55-59, 74, 84, 88, 92, 

122, 128, 139, 144- 
Leibnitz, G. W. F. (16461716). 


Lie, S. (18421899). 1524, 179. 

Liebmann, H. 86, 89, 113, 145, 
180, 220. 

Lindemann, F. 161. 

Lobatschewsky, N. J. (1793 
1856). 44, 51, 55, 63, 65, 74, 
80, 84 99, 1013, 1046, 
1113, Il6 I21 8, 137, 141, 
145, 147, 152, 154, 1578, 
161, 164, 170, 1735, ^778, 

1934, 217, 220, 222, 225, 

238, 249, 2523, 264. 

Lorenz, J. F. (17381807). 58, 


Lukat, M. 129. 

Liitkemeyer, G. 1456. 

Me Cormack, T. J. 182. 

Mach, E. 181. 

Minding, F. (18061885). 132, 


Mobius, A. F. (17901868). 148 


Monge, G. (1746 1818). 545. 
Montucla, J. E. (17251799) 

44- 92- 

Nasir-Eddin, (1201 1274). 10, 
123, i 6 , 378, 120. 

Newton, I. (16421727). 53. 

Olbers,H.W.M.(i758 1840). 65. 

Oliver, (i Half of the 17* Cent 
ury). 17. 

Ovidio, (d ) E. 127. 

Paciolo, Luca (circa 14451514). 


Pascal, E. 127, 139. 

Pasch, M. 176. 

Picard, C. E. 128. 

Poincare, H. 154, 180. 

Poncelet, J. V. (17881867). 155, 

Posidonius, (it Century B.C.). 2, 
3, 8, 14. 

Proclus, (410 4851. 27,123, 
18 20, 119. 

Ptolemy, (87165). 34, 119. 

Riccardi, P. (18281898). 17. 

Ricordi, E. 127. 

Riemann, B. (18261866). 126, 
129, 138-9, 1413, 145 154, 
1578, 1601, 163 4, 175 
_7 ? 179180, 194, 2012. 

Saccheri, G. (16671733). 4, 
224, 26, 2830, 34, 3646, 
51, 55-7, 656, 78, 85, 87 
8, 97, 120, 129, 139, 141, 

Sartorius v. Waltershausen, W. 

(18091876). 122. 
Saville, H. (15491622). 17. 
Schmidt, F. (18261901). 121, 

Schumacher, H. K. (17801850). 

65-7, 75, 122-3, 152. 
Schur, F. H. 176. 
Schweikart, F. K. (17801859). 

67, 75 -78, 80, 83,86, 107, 122. 


Index of Authors. 

Segre, C. 44, 66, 77 8, 92. 

Seyffer, K. F. (17621822). 60, 66. 

Simon, H. 91. 

Simplicius, (6 th Century). 8, 10. 

Sintsoff, D. 139. 

Stackel, P. 16, 445, 50, 60 1, 

63, 66, 823, 101, 1123, 

Staudt,G. C.v.(i798 1867). 129, 

I54> 233, 236. 
Szasz, C. (1798-1853). 97. 
Tannery, P. (18431904). 7, 20. 
Tacquet, A. (16121660). 17. 
Tartaglia, N. (15001557). 17. 
Taurinus, F. A. (17941874). 

656, 74, 779, 813, 87, 

89-91, 94, 99, 112, 137, 173. 

Thibaut, B. F. (17751832). 63. 

Tilly (de), J. M. 55, 114, 194. 

Townsend, E. J. 236. 

Vailati, G. 18, 22. 

Valeric Luca (? 15221618). 17. 

Vasiliev, A. 93, 

Wachter, F. L. (17921817). 62 

3, 66, 88. 
Wallis, J. (1616 1703). 12, 15 

7> 2 9, 53 J20. 
Weber, H. 180. 
Wellstein, J. 180, 238. 
Zamberti, B. (it Half of the 

1 6^ Century). 17. 
Zeno, (495435)- 6 - 
Zolt, A. (de) 127. 

~ ~4T 9**^* I 1 

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Rec d UCflftNr^)s WiJ 1 ftiic 9 < ?nm 

MAR 1 1997 


?rf UCB A/M/< 


OCT 1 2001 


MAV 9 1 , n l MAY o /UU/ 

- Ssp MAY 242003 

NO DD ! 9 BERKELEY, CA 94720